(c) Marcin Sydow
Proofs
Inference Discrete Mathematics rules Rules of Inference and Mathematical Proofs Proofs
Set theory axioms (c) Marcin Sydow Contents
Discrete Mathematics
(c) Marcin Sydow
Proofs
Inference rules Proofs Proofs Rules of inference Set theory axioms Proof types Theorem - a true (proven) mathematical statement. Lemma - a small, helper (technical) theorem. Conjecture - a statement that has not been proven (but is suspected to be true)
Proof
Discrete Mathematics
(c) Marcin Sydow
Proofs A mathematical proof is a (logical) procedure to establish the Inference rules truth of a mathematical statement.
Proofs
Set theory axioms Lemma - a small, helper (technical) theorem. Conjecture - a statement that has not been proven (but is suspected to be true)
Proof
Discrete Mathematics
(c) Marcin Sydow
Proofs A mathematical proof is a (logical) procedure to establish the Inference rules truth of a mathematical statement. Proofs Theorem - a true (proven) mathematical statement. Set theory axioms Conjecture - a statement that has not been proven (but is suspected to be true)
Proof
Discrete Mathematics
(c) Marcin Sydow
Proofs A mathematical proof is a (logical) procedure to establish the Inference rules truth of a mathematical statement. Proofs Theorem - a true (proven) mathematical statement. Set theory axioms Lemma - a small, helper (technical) theorem. Proof
Discrete Mathematics
(c) Marcin Sydow
Proofs A mathematical proof is a (logical) procedure to establish the Inference rules truth of a mathematical statement. Proofs Theorem - a true (proven) mathematical statement. Set theory axioms Lemma - a small, helper (technical) theorem. Conjecture - a statement that has not been proven (but is suspected to be true) A formal proof of the conclusion C based on the set of premises and axioms P is a sequence S = {S1, S2, ..., Sn} of logical statements so that each statement Si is either:
a premise or axiom from the set P a tautology a subconclusion derived from (some of) the previous statements Sk , k < i in the sequence using some of the allowed inference rules or substitution rules.
Formal proof
Discrete Mathematics
(c) Marcin Sydow Let P = {P1, P2, ..., Pm} be a set of premises or axioms and let C be a conclusion do be proven. Proofs
Inference rules
Proofs
Set theory axioms a premise or axiom from the set P a tautology a subconclusion derived from (some of) the previous statements Sk , k < i in the sequence using some of the allowed inference rules or substitution rules.
Formal proof
Discrete Mathematics
(c) Marcin Sydow Let P = {P1, P2, ..., Pm} be a set of premises or axioms and let C be a conclusion do be proven. Proofs
Inference A formal proof of the conclusion C based on the set of rules premises and axioms P is a sequence S = {S1, S2, ..., Sn} of Proofs logical statements so that each statement Si is either: Set theory axioms Formal proof
Discrete Mathematics
(c) Marcin Sydow Let P = {P1, P2, ..., Pm} be a set of premises or axioms and let C be a conclusion do be proven. Proofs
Inference A formal proof of the conclusion C based on the set of rules premises and axioms P is a sequence S = {S1, S2, ..., Sn} of Proofs logical statements so that each statement Si is either: Set theory axioms a premise or axiom from the set P a tautology a subconclusion derived from (some of) the previous statements Sk , k < i in the sequence using some of the allowed inference rules or substitution rules. If a compound proposition P is a tautology and contains another proposition Q and all the occurrences of Q are substituted with another proposition Q∗ that is logically equivalent to Q, then the resulting compound proposition is also a tautology.
Substition rules
Discrete Mathematics
(c) Marcin The following rules make it possible to build new tautologies Sydow out of the existing ones. Proofs
Inference If a compound proposition P is a tautology and all the rules occurrences of some speci c variable of P are substituted Proofs with the same proposition E, then the resulting compound Set theory axioms proposition is also a tautology. Substition rules
Discrete Mathematics
(c) Marcin The following rules make it possible to build new tautologies Sydow out of the existing ones. Proofs
Inference If a compound proposition P is a tautology and all the rules occurrences of some speci c variable of P are substituted Proofs with the same proposition E, then the resulting compound Set theory axioms proposition is also a tautology. If a compound proposition P is a tautology and contains another proposition Q and all the occurrences of Q are substituted with another proposition Q∗ that is logically equivalent to Q, then the resulting compound proposition is also a tautology. p [(p) ∧ (q)] → (p ∧ q) conjunction q ∴ p ∧ q p p → (p ∨ q) addition ∴ p ∨ q p ∨ q [(p ∨ q) ∧ (¬p ∨ r)] → (q ∨ r) resolution ¬p ∨ r ∴ q ∨ r (to be continued on the next slide)
Inference rules 1
Discrete Mathematics The following rules make it possible to derive next steps of a proof (c) Marcin based on the previous steps or premises and axioms: Sydow
Proofs Rule of inference Tautology Name
Inference p ∧ q (p ∧ q) → p simpli cation rules ∴ p Proofs
Set theory axioms p p → (p ∨ q) addition ∴ p ∨ q p ∨ q [(p ∨ q) ∧ (¬p ∨ r)] → (q ∨ r) resolution ¬p ∨ r ∴ q ∨ r (to be continued on the next slide)
Inference rules 1
Discrete Mathematics The following rules make it possible to derive next steps of a proof (c) Marcin based on the previous steps or premises and axioms: Sydow
Proofs Rule of inference Tautology Name
Inference p ∧ q (p ∧ q) → p simpli cation rules ∴ p Proofs p [(p) ∧ (q)] → (p ∧ q) conjunction Set theory q axioms ∴ p ∧ q p ∨ q [(p ∨ q) ∧ (¬p ∨ r)] → (q ∨ r) resolution ¬p ∨ r ∴ q ∨ r (to be continued on the next slide)
Inference rules 1
Discrete Mathematics The following rules make it possible to derive next steps of a proof (c) Marcin based on the previous steps or premises and axioms: Sydow
Proofs Rule of inference Tautology Name
Inference p ∧ q (p ∧ q) → p simpli cation rules ∴ p Proofs p [(p) ∧ (q)] → (p ∧ q) conjunction Set theory q axioms ∴ p ∧ q p p → (p ∨ q) addition ∴ p ∨ q Inference rules 1
Discrete Mathematics The following rules make it possible to derive next steps of a proof (c) Marcin based on the previous steps or premises and axioms: Sydow
Proofs Rule of inference Tautology Name
Inference p ∧ q (p ∧ q) → p simpli cation rules ∴ p Proofs p [(p) ∧ (q)] → (p ∧ q) conjunction Set theory q axioms ∴ p ∧ q p p → (p ∨ q) addition ∴ p ∨ q p ∨ q [(p ∨ q) ∧ (¬p ∨ r)] → (q ∨ r) resolution ¬p ∨ r ∴ q ∨ r (to be continued on the next slide) ¬q [¬q ∧ (p → q)] → ¬p Modus tollens p → q ∴ ¬p p → q [(p → q) ∧ (q → r)] → (p → r) Hypothetical q → r syllogism ∴ p → q p ∨ q [(p ∨ q) ∧ ¬p] → q Disjunctive ¬p syllogism ∴ q
Inference rules 2
Discrete Mathematics
(c) Marcin Sydow Rule of inference Tautology Name p [p ∧ (p → q)] → q Modus ponens Proofs p → q Inference q rules ∴
Proofs
Set theory axioms p → q [(p → q) ∧ (q → r)] → (p → r) Hypothetical q → r syllogism ∴ p → q p ∨ q [(p ∨ q) ∧ ¬p] → q Disjunctive ¬p syllogism ∴ q
Inference rules 2
Discrete Mathematics
(c) Marcin Sydow Rule of inference Tautology Name p [p ∧ (p → q)] → q Modus ponens Proofs p → q Inference q rules ∴ ¬q [¬q ∧ (p → q)] → ¬p Modus tollens Proofs p → q Set theory ¬p axioms ∴ p ∨ q [(p ∨ q) ∧ ¬p] → q Disjunctive ¬p syllogism ∴ q
Inference rules 2
Discrete Mathematics
(c) Marcin Sydow Rule of inference Tautology Name p [p ∧ (p → q)] → q Modus ponens Proofs p → q Inference q rules ∴ ¬q [¬q ∧ (p → q)] → ¬p Modus tollens Proofs p → q Set theory ¬p axioms ∴ p → q [(p → q) ∧ (q → r)] → (p → r) Hypothetical q → r syllogism ∴ p → q Inference rules 2
Discrete Mathematics
(c) Marcin Sydow Rule of inference Tautology Name p [p ∧ (p → q)] → q Modus ponens Proofs p → q Inference q rules ∴ ¬q [¬q ∧ (p → q)] → ¬p Modus tollens Proofs p → q Set theory ¬p axioms ∴ p → q [(p → q) ∧ (q → r)] → (p → r) Hypothetical q → r syllogism ∴ p → q p ∨ q [(p ∨ q) ∧ ¬p] → q Disjunctive ¬p syllogism ∴ q P(c) for an arbitrary c Universal generalization ∴ ∀x P(x)
∃x P(x) Existential instantiation ∴ P(c) for some element c
P(c) for some element c Existential generalization ∴ ∃x P(x)
Inference rules for quanti ed predicates
Discrete Rule of inference Name Mathematics
(c) Marcin Sydow ∀x P(x) Universal instantiation ∴ P(c) Proofs
Inference rules
Proofs
Set theory axioms ∃x P(x) Existential instantiation ∴ P(c) for some element c
P(c) for some element c Existential generalization ∴ ∃x P(x)
Inference rules for quanti ed predicates
Discrete Rule of inference Name Mathematics
(c) Marcin Sydow ∀x P(x) Universal instantiation ∴ P(c) Proofs
Inference rules P c for an arbitrary c Universal generalization Proofs ( ) P x Set theory ∴ ∀x ( ) axioms P(c) for some element c Existential generalization ∴ ∃x P(x)
Inference rules for quanti ed predicates
Discrete Rule of inference Name Mathematics
(c) Marcin Sydow ∀x P(x) Universal instantiation ∴ P(c) Proofs
Inference rules P c for an arbitrary c Universal generalization Proofs ( ) P x Set theory ∴ ∀x ( ) axioms
∃x P(x) Existential instantiation ∴ P(c) for some element c Inference rules for quanti ed predicates
Discrete Rule of inference Name Mathematics
(c) Marcin Sydow ∀x P(x) Universal instantiation ∴ P(c) Proofs
Inference rules P c for an arbitrary c Universal generalization Proofs ( ) P x Set theory ∴ ∀x ( ) axioms
∃x P(x) Existential instantiation ∴ P(c) for some element c
P(c) for some element c Existential generalization ∴ ∃x P(x) The proof can have various forms, e.g.: direct proof (using P to directly show C) indirect proof proof by contraposition (proving contrapostion ¬C ⇒ ¬P proof by contradiction (reductio ad absurdum) (showing that P ∧ ¬C leads to false (absurd)) Another proof scheme is proof by cases (when dierent cases are treated separately).
Types of proof of implication
Discrete Mathematics Assume that theorem is of the form:
(c) Marcin Sydow P ⇒ C Proofs (where P P P P is the conjunction of premises and Inference = 1 ∧ 2 ∧ ... m rules axioms, and C is the conclusion to be proven) Proofs
Set theory axioms direct proof (using P to directly show C) indirect proof proof by contraposition (proving contrapostion ¬C ⇒ ¬P proof by contradiction (reductio ad absurdum) (showing that P ∧ ¬C leads to false (absurd)) Another proof scheme is proof by cases (when dierent cases are treated separately).
Types of proof of implication
Discrete Mathematics Assume that theorem is of the form:
(c) Marcin Sydow P ⇒ C Proofs (where P P P P is the conjunction of premises and Inference = 1 ∧ 2 ∧ ... m rules axioms, and C is the conclusion to be proven) Proofs
Set theory The proof can have various forms, e.g.: axioms indirect proof proof by contraposition (proving contrapostion ¬C ⇒ ¬P proof by contradiction (reductio ad absurdum) (showing that P ∧ ¬C leads to false (absurd)) Another proof scheme is proof by cases (when dierent cases are treated separately).
Types of proof of implication
Discrete Mathematics Assume that theorem is of the form:
(c) Marcin Sydow P ⇒ C Proofs (where P P P P is the conjunction of premises and Inference = 1 ∧ 2 ∧ ... m rules axioms, and C is the conclusion to be proven) Proofs
Set theory The proof can have various forms, e.g.: axioms direct proof (using P to directly show C) proof by contraposition (proving contrapostion ¬C ⇒ ¬P proof by contradiction (reductio ad absurdum) (showing that P ∧ ¬C leads to false (absurd)) Another proof scheme is proof by cases (when dierent cases are treated separately).
Types of proof of implication
Discrete Mathematics Assume that theorem is of the form:
(c) Marcin Sydow P ⇒ C Proofs (where P P P P is the conjunction of premises and Inference = 1 ∧ 2 ∧ ... m rules axioms, and C is the conclusion to be proven) Proofs
Set theory The proof can have various forms, e.g.: axioms direct proof (using P to directly show C) indirect proof proof by contradiction (reductio ad absurdum) (showing that P ∧ ¬C leads to false (absurd)) Another proof scheme is proof by cases (when dierent cases are treated separately).
Types of proof of implication
Discrete Mathematics Assume that theorem is of the form:
(c) Marcin Sydow P ⇒ C Proofs (where P P P P is the conjunction of premises and Inference = 1 ∧ 2 ∧ ... m rules axioms, and C is the conclusion to be proven) Proofs
Set theory The proof can have various forms, e.g.: axioms direct proof (using P to directly show C) indirect proof proof by contraposition (proving contrapostion ¬C ⇒ ¬P Another proof scheme is proof by cases (when dierent cases are treated separately).
Types of proof of implication
Discrete Mathematics Assume that theorem is of the form:
(c) Marcin Sydow P ⇒ C Proofs (where P P P P is the conjunction of premises and Inference = 1 ∧ 2 ∧ ... m rules axioms, and C is the conclusion to be proven) Proofs
Set theory The proof can have various forms, e.g.: axioms direct proof (using P to directly show C) indirect proof proof by contraposition (proving contrapostion ¬C ⇒ ¬P proof by contradiction (reductio ad absurdum) (showing that P ∧ ¬C leads to false (absurd)) Types of proof of implication
Discrete Mathematics Assume that theorem is of the form:
(c) Marcin Sydow P ⇒ C Proofs (where P P P P is the conjunction of premises and Inference = 1 ∧ 2 ∧ ... m rules axioms, and C is the conclusion to be proven) Proofs
Set theory The proof can have various forms, e.g.: axioms direct proof (using P to directly show C) indirect proof proof by contraposition (proving contrapostion ¬C ⇒ ¬P proof by contradiction (reductio ad absurdum) (showing that P ∧ ¬C leads to false (absurd)) Another proof scheme is proof by cases (when dierent cases are treated separately). (actually more formally it is: ∀n ∈ Z(∃k ∈ Z n = (2k + 1)) → (∃m ∈ Z n2 = (2m + 1))) n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1 (thus m = (2k2 + 2k)) Another example: if m and n are squares then mn is square
Example of a direct proof
Discrete Mathematics
(c) Marcin Sydow
Proofs Theorem: if n is odd integer then n2 is odd. Inference (what is the mathematical form of the above statement?) rules
Proofs
Set theory axioms n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1 (thus m = (2k2 + 2k)) Another example: if m and n are squares then mn is square
Example of a direct proof
Discrete Mathematics
(c) Marcin Sydow
Proofs Theorem: if n is odd integer then n2 is odd. Inference (what is the mathematical form of the above statement?) rules
Proofs (actually more formally it is: 2 Set theory ∀n ∈ Z(∃k ∈ Z n = (2k + 1)) → (∃m ∈ Z n = (2m + 1))) axioms Another example: if m and n are squares then mn is square
Example of a direct proof
Discrete Mathematics
(c) Marcin Sydow
Proofs Theorem: if n is odd integer then n2 is odd. Inference (what is the mathematical form of the above statement?) rules
Proofs (actually more formally it is: 2 Set theory ∀n ∈ Z(∃k ∈ Z n = (2k + 1)) → (∃m ∈ Z n = (2m + 1))) axioms n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1 (thus m = (2k2 + 2k)) Example of a direct proof
Discrete Mathematics
(c) Marcin Sydow
Proofs Theorem: if n is odd integer then n2 is odd. Inference (what is the mathematical form of the above statement?) rules
Proofs (actually more formally it is: 2 Set theory ∀n ∈ Z(∃k ∈ Z n = (2k + 1)) → (∃m ∈ Z n = (2m + 1))) axioms n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1 (thus m = (2k2 + 2k)) Another example: if m and n are squares then mn is square x is rational if there exist two integers p,q so that x = p/q (it is easy to use basic algebra to show that x + y is also rational)
Example of direct proof
Discrete Mathematics
(c) Marcin Sydow
Proofs
Inference rules Sum of two rationals is rational Proofs
Set theory axioms (it is easy to use basic algebra to show that x + y is also rational)
Example of direct proof
Discrete Mathematics
(c) Marcin Sydow
Proofs
Inference rules Sum of two rationals is rational Proofs x is rational if there exist two integers p,q so that x = p/q Set theory axioms Example of direct proof
Discrete Mathematics
(c) Marcin Sydow
Proofs
Inference rules Sum of two rationals is rational Proofs x is rational if there exist two integers p,q so that x = p/q Set theory (it is easy to use basic algebra to show that x y is also axioms + rational) (p → q) ⇔ (¬p ∨ q) implication as alternative (p → q) ⇔ ¬(p ∧ ¬q) implication as conjuction [p → (q ∧ r)] ⇔ [(p → q) ∧ (p → r)] splitting a conjunction (p → q) ⇔ [(p ∧ ¬q) → F ] reductio ad absurdum [(p ∧ q) → r] ⇔ [p → (q → r)] exportation law (p ↔ q) ⇔ [(p → q) ∧ (q → p)] bidirectional as implications The last identity gives a schema for proving equivalences. The above identities serve as a basis for various types of proofs, e.g.: indirect proof by contraposition (by proving the negation of the premise from the negation of the conclusion) indirect vacuous proof (by observing that the premise is false) indirect trivial proof (by ignoring the premise) indirect proof by contradiction (by showing that the negation of the conclusion leads to a contradiction)
Logical identities useful in proving implications
Discrete Identity: Name: Mathematics (p → q) ⇔ (¬q → ¬p) contraposition (c) Marcin Sydow
Proofs
Inference rules
Proofs
Set theory axioms (p → q) ⇔ ¬(p ∧ ¬q) implication as conjuction [p → (q ∧ r)] ⇔ [(p → q) ∧ (p → r)] splitting a conjunction (p → q) ⇔ [(p ∧ ¬q) → F ] reductio ad absurdum [(p ∧ q) → r] ⇔ [p → (q → r)] exportation law (p ↔ q) ⇔ [(p → q) ∧ (q → p)] bidirectional as implications The last identity gives a schema for proving equivalences. The above identities serve as a basis for various types of proofs, e.g.: indirect proof by contraposition (by proving the negation of the premise from the negation of the conclusion) indirect vacuous proof (by observing that the premise is false) indirect trivial proof (by ignoring the premise) indirect proof by contradiction (by showing that the negation of the conclusion leads to a contradiction)
Logical identities useful in proving implications
Discrete Identity: Name: Mathematics (p → q) ⇔ (¬q → ¬p) contraposition (c) Marcin (p → q) ⇔ (¬p ∨ q) implication as alternative Sydow
Proofs
Inference rules
Proofs
Set theory axioms [p → (q ∧ r)] ⇔ [(p → q) ∧ (p → r)] splitting a conjunction (p → q) ⇔ [(p ∧ ¬q) → F ] reductio ad absurdum [(p ∧ q) → r] ⇔ [p → (q → r)] exportation law (p ↔ q) ⇔ [(p → q) ∧ (q → p)] bidirectional as implications The last identity gives a schema for proving equivalences. The above identities serve as a basis for various types of proofs, e.g.: indirect proof by contraposition (by proving the negation of the premise from the negation of the conclusion) indirect vacuous proof (by observing that the premise is false) indirect trivial proof (by ignoring the premise) indirect proof by contradiction (by showing that the negation of the conclusion leads to a contradiction)
Logical identities useful in proving implications
Discrete Identity: Name: Mathematics (p → q) ⇔ (¬q → ¬p) contraposition (c) Marcin (p → q) ⇔ (¬p ∨ q) implication as alternative Sydow (p → q) ⇔ ¬(p ∧ ¬q) implication as conjuction Proofs
Inference rules
Proofs
Set theory axioms (p → q) ⇔ [(p ∧ ¬q) → F ] reductio ad absurdum [(p ∧ q) → r] ⇔ [p → (q → r)] exportation law (p ↔ q) ⇔ [(p → q) ∧ (q → p)] bidirectional as implications The last identity gives a schema for proving equivalences. The above identities serve as a basis for various types of proofs, e.g.: indirect proof by contraposition (by proving the negation of the premise from the negation of the conclusion) indirect vacuous proof (by observing that the premise is false) indirect trivial proof (by ignoring the premise) indirect proof by contradiction (by showing that the negation of the conclusion leads to a contradiction)
Logical identities useful in proving implications
Discrete Identity: Name: Mathematics (p → q) ⇔ (¬q → ¬p) contraposition (c) Marcin (p → q) ⇔ (¬p ∨ q) implication as alternative Sydow (p → q) ⇔ ¬(p ∧ ¬q) implication as conjuction Proofs [p → (q ∧ r)] ⇔ [(p → q) ∧ (p → r)] splitting a conjunction Inference rules
Proofs
Set theory axioms [(p ∧ q) → r] ⇔ [p → (q → r)] exportation law (p ↔ q) ⇔ [(p → q) ∧ (q → p)] bidirectional as implications The last identity gives a schema for proving equivalences. The above identities serve as a basis for various types of proofs, e.g.: indirect proof by contraposition (by proving the negation of the premise from the negation of the conclusion) indirect vacuous proof (by observing that the premise is false) indirect trivial proof (by ignoring the premise) indirect proof by contradiction (by showing that the negation of the conclusion leads to a contradiction)
Logical identities useful in proving implications
Discrete Identity: Name: Mathematics (p → q) ⇔ (¬q → ¬p) contraposition (c) Marcin (p → q) ⇔ (¬p ∨ q) implication as alternative Sydow (p → q) ⇔ ¬(p ∧ ¬q) implication as conjuction Proofs [p → (q ∧ r)] ⇔ [(p → q) ∧ (p → r)] splitting a conjunction Inference (p → q) ⇔ [(p ∧ ¬q) → F ] reductio ad absurdum rules
Proofs
Set theory axioms (p ↔ q) ⇔ [(p → q) ∧ (q → p)] bidirectional as implications The last identity gives a schema for proving equivalences. The above identities serve as a basis for various types of proofs, e.g.: indirect proof by contraposition (by proving the negation of the premise from the negation of the conclusion) indirect vacuous proof (by observing that the premise is false) indirect trivial proof (by ignoring the premise) indirect proof by contradiction (by showing that the negation of the conclusion leads to a contradiction)
Logical identities useful in proving implications
Discrete Identity: Name: Mathematics (p → q) ⇔ (¬q → ¬p) contraposition (c) Marcin (p → q) ⇔ (¬p ∨ q) implication as alternative Sydow (p → q) ⇔ ¬(p ∧ ¬q) implication as conjuction Proofs [p → (q ∧ r)] ⇔ [(p → q) ∧ (p → r)] splitting a conjunction Inference (p → q) ⇔ [(p ∧ ¬q) → F ] reductio ad absurdum rules [(p ∧ q) → r] ⇔ [p → (q → r)] exportation law Proofs
Set theory axioms The above identities serve as a basis for various types of proofs, e.g.: indirect proof by contraposition (by proving the negation of the premise from the negation of the conclusion) indirect vacuous proof (by observing that the premise is false) indirect trivial proof (by ignoring the premise) indirect proof by contradiction (by showing that the negation of the conclusion leads to a contradiction)
Logical identities useful in proving implications
Discrete Identity: Name: Mathematics (p → q) ⇔ (¬q → ¬p) contraposition (c) Marcin (p → q) ⇔ (¬p ∨ q) implication as alternative Sydow (p → q) ⇔ ¬(p ∧ ¬q) implication as conjuction Proofs [p → (q ∧ r)] ⇔ [(p → q) ∧ (p → r)] splitting a conjunction Inference (p → q) ⇔ [(p ∧ ¬q) → F ] reductio ad absurdum rules [(p ∧ q) → r] ⇔ [p → (q → r)] exportation law Proofs (p ↔ q) ⇔ [(p → q) ∧ (q → p)] bidirectional as implications Set theory The last identity gives a schema for proving equivalences. axioms indirect vacuous proof (by observing that the premise is false) indirect trivial proof (by ignoring the premise) indirect proof by contradiction (by showing that the negation of the conclusion leads to a contradiction)
Logical identities useful in proving implications
Discrete Identity: Name: Mathematics (p → q) ⇔ (¬q → ¬p) contraposition (c) Marcin (p → q) ⇔ (¬p ∨ q) implication as alternative Sydow (p → q) ⇔ ¬(p ∧ ¬q) implication as conjuction Proofs [p → (q ∧ r)] ⇔ [(p → q) ∧ (p → r)] splitting a conjunction Inference (p → q) ⇔ [(p ∧ ¬q) → F ] reductio ad absurdum rules [(p ∧ q) → r] ⇔ [p → (q → r)] exportation law Proofs (p ↔ q) ⇔ [(p → q) ∧ (q → p)] bidirectional as implications Set theory The last identity gives a schema for proving equivalences. axioms The above identities serve as a basis for various types of proofs, e.g.: indirect proof by contraposition (by proving the negation of the premise from the negation of the conclusion) indirect trivial proof (by ignoring the premise) indirect proof by contradiction (by showing that the negation of the conclusion leads to a contradiction)
Logical identities useful in proving implications
Discrete Identity: Name: Mathematics (p → q) ⇔ (¬q → ¬p) contraposition (c) Marcin (p → q) ⇔ (¬p ∨ q) implication as alternative Sydow (p → q) ⇔ ¬(p ∧ ¬q) implication as conjuction Proofs [p → (q ∧ r)] ⇔ [(p → q) ∧ (p → r)] splitting a conjunction Inference (p → q) ⇔ [(p ∧ ¬q) → F ] reductio ad absurdum rules [(p ∧ q) → r] ⇔ [p → (q → r)] exportation law Proofs (p ↔ q) ⇔ [(p → q) ∧ (q → p)] bidirectional as implications Set theory The last identity gives a schema for proving equivalences. axioms The above identities serve as a basis for various types of proofs, e.g.: indirect proof by contraposition (by proving the negation of the premise from the negation of the conclusion) indirect vacuous proof (by observing that the premise is false) indirect proof by contradiction (by showing that the negation of the conclusion leads to a contradiction)
Logical identities useful in proving implications
Discrete Identity: Name: Mathematics (p → q) ⇔ (¬q → ¬p) contraposition (c) Marcin (p → q) ⇔ (¬p ∨ q) implication as alternative Sydow (p → q) ⇔ ¬(p ∧ ¬q) implication as conjuction Proofs [p → (q ∧ r)] ⇔ [(p → q) ∧ (p → r)] splitting a conjunction Inference (p → q) ⇔ [(p ∧ ¬q) → F ] reductio ad absurdum rules [(p ∧ q) → r] ⇔ [p → (q → r)] exportation law Proofs (p ↔ q) ⇔ [(p → q) ∧ (q → p)] bidirectional as implications Set theory The last identity gives a schema for proving equivalences. axioms The above identities serve as a basis for various types of proofs, e.g.: indirect proof by contraposition (by proving the negation of the premise from the negation of the conclusion) indirect vacuous proof (by observing that the premise is false) indirect trivial proof (by ignoring the premise) Logical identities useful in proving implications
Discrete Identity: Name: Mathematics (p → q) ⇔ (¬q → ¬p) contraposition (c) Marcin (p → q) ⇔ (¬p ∨ q) implication as alternative Sydow (p → q) ⇔ ¬(p ∧ ¬q) implication as conjuction Proofs [p → (q ∧ r)] ⇔ [(p → q) ∧ (p → r)] splitting a conjunction Inference (p → q) ⇔ [(p ∧ ¬q) → F ] reductio ad absurdum rules [(p ∧ q) → r] ⇔ [p → (q → r)] exportation law Proofs (p ↔ q) ⇔ [(p → q) ∧ (q → p)] bidirectional as implications Set theory The last identity gives a schema for proving equivalences. axioms The above identities serve as a basis for various types of proofs, e.g.: indirect proof by contraposition (by proving the negation of the premise from the negation of the conclusion) indirect vacuous proof (by observing that the premise is false) indirect trivial proof (by ignoring the premise) indirect proof by contradiction (by showing that the negation of the conclusion leads to a contradiction) (how to prove it with a direct proof?) (it is not easy to construct a direct proof, but an indirect proof can be easily presented)
Example of the need for indirect proofs
Discrete Mathematics
(c) Marcin Sydow
Proofs
Inference rules Prove: for any integer n: if 3n+2 is odd then n is odd Proofs
Set theory axioms (it is not easy to construct a direct proof, but an indirect proof can be easily presented)
Example of the need for indirect proofs
Discrete Mathematics
(c) Marcin Sydow
Proofs
Inference rules Prove: for any integer n: if 3n+2 is odd then n is odd Proofs (how to prove it with a direct proof?) Set theory axioms Example of the need for indirect proofs
Discrete Mathematics
(c) Marcin Sydow
Proofs
Inference rules Prove: for any integer n: if 3n+2 is odd then n is odd Proofs (how to prove it with a direct proof?) Set theory (it is not easy to construct a direct proof, but an indirect proof axioms can be easily presented) (by contraposition): Assume n is even: ∃k ∈ Z n = 2k, which implies: 3n + 2 = 3(2k) + 2 = 2(3k) + 2 = 2(3k + 1) = 2(l) (where l = 3k + 1) what would imply that the number 3n + 2 is also an even number (contraposition)
Example of a proof by contraposition
Discrete Mathematics
(c) Marcin Sydow
Proofs Prove: for any integer n: if 3n+2 is odd then n is odd Inference rules (example of indirect proof): Proofs
Set theory axioms Assume n is even: ∃k ∈ Z n = 2k, which implies: 3n + 2 = 3(2k) + 2 = 2(3k) + 2 = 2(3k + 1) = 2(l) (where l = 3k + 1) what would imply that the number 3n + 2 is also an even number (contraposition)
Example of a proof by contraposition
Discrete Mathematics
(c) Marcin Sydow
Proofs Prove: for any integer n: if 3n+2 is odd then n is odd Inference rules (example of indirect proof): Proofs (by contraposition): Set theory axioms Example of a proof by contraposition
Discrete Mathematics
(c) Marcin Sydow
Proofs Prove: for any integer n: if 3n+2 is odd then n is odd Inference rules (example of indirect proof): Proofs (by contraposition): Assume n is even: ∃k ∈ Z n = 2k, which Set theory axioms implies: 3n + 2 = 3(2k) + 2 = 2(3k) + 2 = 2(3k + 1) = 2(l) (where l = 3k + 1) what would imply that the number 3n + 2 is also an even number (contraposition) de ne a predicate P(n): if n > 1 then n2 > n (n ∈ Z) Prove P(0). The hypothesis n > 1 is false so the implication is automatically true. Vacuous proofs are useful for example for proving the base step in mathematical induction1
Example of a vacuous proof
Discrete Mathematics
(c) Marcin Sydow
Proofs (when the hypothesis of the implication is false) Inference rules
Proofs
Set theory axioms
1a proof technique that will be presented later Prove P(0). The hypothesis n > 1 is false so the implication is automatically true. Vacuous proofs are useful for example for proving the base step in mathematical induction1
Example of a vacuous proof
Discrete Mathematics
(c) Marcin Sydow
Proofs (when the hypothesis of the implication is false) Inference de ne a predicate P(n): if n 1 then n2 n (n Z) rules > > ∈
Proofs
Set theory axioms
1a proof technique that will be presented later The hypothesis n > 1 is false so the implication is automatically true. Vacuous proofs are useful for example for proving the base step in mathematical induction1
Example of a vacuous proof
Discrete Mathematics
(c) Marcin Sydow
Proofs (when the hypothesis of the implication is false) Inference de ne a predicate P(n): if n 1 then n2 n (n Z) rules > > ∈ Prove P(0). Proofs
Set theory axioms
1a proof technique that will be presented later Vacuous proofs are useful for example for proving the base step in mathematical induction1
Example of a vacuous proof
Discrete Mathematics
(c) Marcin Sydow
Proofs (when the hypothesis of the implication is false) Inference de ne a predicate P(n): if n 1 then n2 n (n Z) rules > > ∈ Prove P(0). Proofs
Set theory The hypothesis n > 1 is false so the implication is automatically axioms true.
1a proof technique that will be presented later Example of a vacuous proof
Discrete Mathematics
(c) Marcin Sydow
Proofs (when the hypothesis of the implication is false) Inference de ne a predicate P(n): if n 1 then n2 n (n Z) rules > > ∈ Prove P(0). Proofs
Set theory The hypothesis n > 1 is false so the implication is automatically axioms true. Vacuous proofs are useful for example for proving the base step in mathematical induction1
1a proof technique that will be presented later de ne the predicate: P(n): for all positive integers a,b and natural number n it holds that: a ≥ b ⇒ an ≥ bn. Prove P(0) a0 = 1 = b0 so that the conclusion is true without the hypothesis assumption
An example of a trivial proof
Discrete Mathematics
(c) Marcin Sydow
Proofs
Inference (when the the hypothesis of the implication can be ignored) rules
Proofs
Set theory axioms Prove P(0) a0 = 1 = b0 so that the conclusion is true without the hypothesis assumption
An example of a trivial proof
Discrete Mathematics
(c) Marcin Sydow
Proofs
Inference (when the the hypothesis of the implication can be ignored) rules de ne the predicate: P(n): for all positive integers a,b and Proofs natural number n it holds that: a ≥ b ⇒ an ≥ bn. Set theory axioms a0 = 1 = b0 so that the conclusion is true without the hypothesis assumption
An example of a trivial proof
Discrete Mathematics
(c) Marcin Sydow
Proofs
Inference (when the the hypothesis of the implication can be ignored) rules de ne the predicate: P(n): for all positive integers a,b and Proofs natural number n it holds that: a ≥ b ⇒ an ≥ bn. Set theory axioms Prove P(0) An example of a trivial proof
Discrete Mathematics
(c) Marcin Sydow
Proofs
Inference (when the the hypothesis of the implication can be ignored) rules de ne the predicate: P(n): for all positive integers a,b and Proofs natural number n it holds that: a ≥ b ⇒ an ≥ bn. Set theory axioms Prove P(0) a0 = 1 = b0 so that the conclusion is true without the hypothesis assumption (we use the fact that each natural n > 1 is a unique product of prime numbers) √ Suppose that it is not true, i.e. 2 = a/b for some a, b ∈ Z and a, b have no common factors (except 1). 2 = a2/b2 so 2b2 = a2, so a2 is even (divisible by 2). But this implies that b must also be divisible by 2, what contradicts the assumption. Thus negating the thesis leads to a contradiction.
Example of a proof by contradiction
Discrete Mathematics
(c) Marcin Sydow √ 2 is irrational Proofs
Inference rules
Proofs
Set theory axioms 2 = a2/b2 so 2b2 = a2, so a2 is even (divisible by 2). But this implies that b must also be divisible by 2, what contradicts the assumption. Thus negating the thesis leads to a contradiction.
Example of a proof by contradiction
Discrete Mathematics
(c) Marcin Sydow √ 2 is irrational Proofs (we use the fact that each natural n > 1 is a unique product of Inference rules prime numbers) √ Proofs Suppose that it is not true, i.e. 2 = a/b for some a, b ∈ Z Set theory and a b have no common factors (except 1). axioms , Thus negating the thesis leads to a contradiction.
Example of a proof by contradiction
Discrete Mathematics
(c) Marcin Sydow √ 2 is irrational Proofs (we use the fact that each natural n > 1 is a unique product of Inference rules prime numbers) √ Proofs Suppose that it is not true, i.e. 2 = a/b for some a, b ∈ Z Set theory and a b have no common factors (except 1). axioms , 2 = a2/b2 so 2b2 = a2, so a2 is even (divisible by 2). But this implies that b must also be divisible by 2, what contradicts the assumption. Example of a proof by contradiction
Discrete Mathematics
(c) Marcin Sydow √ 2 is irrational Proofs (we use the fact that each natural n > 1 is a unique product of Inference rules prime numbers) √ Proofs Suppose that it is not true, i.e. 2 = a/b for some a, b ∈ Z Set theory and a b have no common factors (except 1). axioms , 2 = a2/b2 so 2b2 = a2, so a2 is even (divisible by 2). But this implies that b must also be divisible by 2, what contradicts the assumption. Thus negating the thesis leads to a contradiction. constructive (by directly presenting an object having the properties or presenting a sure way in which such object can be constructed) unconstructive (without constructing or presenting the object)
Proofs of existential statements
Discrete Mathematics
(c) Marcin Sydow
Proofs If the conclusion is of the form there exists some object that Inference has some properties (∃), the proof can be: rules
Proofs
Set theory axioms unconstructive (without constructing or presenting the object)
Proofs of existential statements
Discrete Mathematics
(c) Marcin Sydow
Proofs If the conclusion is of the form there exists some object that Inference has some properties (∃), the proof can be: rules constructive (by directly presenting an object having the Proofs
Set theory properties or presenting a sure way in which such object axioms can be constructed) Proofs of existential statements
Discrete Mathematics
(c) Marcin Sydow
Proofs If the conclusion is of the form there exists some object that Inference has some properties (∃), the proof can be: rules constructive (by directly presenting an object having the Proofs
Set theory properties or presenting a sure way in which such object axioms can be constructed) unconstructive (without constructing or presenting the object) Example of a constructive proof
Discrete Mathematics
(c) Marcin Sydow
Proofs
Inference rules There exists pair of rational numbers x,y so that xy is Proofs irrational Set theory axioms Proof (constructive): x = 2, y = 1/2 √ Proof: (use the premise that 2 is irrational that was proven √ √ before) Let's de ne x 2 2. If x is rational, this ends the = √ proof. If x is irrational, then x 2 = 2 so that we found another pair. Notice: we do not know which case it true, but we've proven that at least one pair must exist!
Example of a non-constructive proof
Discrete Mathematics
(c) Marcin Sydow
y Proofs There exist irrational numbers x and y so that x is rational. Inference rules
Proofs
Set theory axioms Notice: we do not know which case it true, but we've proven that at least one pair must exist!
Example of a non-constructive proof
Discrete Mathematics
(c) Marcin Sydow
y Proofs There exist irrational numbers x and y so that x is rational. √ Inference Proof: (use the premise that 2 is irrational that was proven rules √ √ 2 Proofs before) Let's de ne x 2 . If x is rational, this ends the = √ Set theory proof. If x is irrational, then x 2 2 so that we found another axioms = pair. Example of a non-constructive proof
Discrete Mathematics
(c) Marcin Sydow
y Proofs There exist irrational numbers x and y so that x is rational. √ Inference Proof: (use the premise that 2 is irrational that was proven rules √ √ 2 Proofs before) Let's de ne x 2 . If x is rational, this ends the = √ Set theory proof. If x is irrational, then x 2 2 so that we found another axioms = pair. Notice: we do not know which case it true, but we've proven that at least one pair must exist! To make a positive proof of a universal statement, if the domain is in nite, it is not possible to prove it for all cases. Instead, the negation of it can be falsi ed, for example.
Proofs of universal statements
Discrete Mathematics
(c) Marcin Sydow
Proofs If the conclusion to be proven starts with the universal Inference quanti er , we can disprove it (prove it is false) by nding a rules ∀
Proofs counterexample (it is an allowed value of the quanti ed
Set theory variable that falsi es the statement). axioms Proofs of universal statements
Discrete Mathematics
(c) Marcin Sydow
Proofs If the conclusion to be proven starts with the universal Inference quanti er , we can disprove it (prove it is false) by nding a rules ∀
Proofs counterexample (it is an allowed value of the quanti ed
Set theory variable that falsi es the statement). axioms To make a positive proof of a universal statement, if the domain is in nite, it is not possible to prove it for all cases. Instead, the negation of it can be falsi ed, for example. A typical proof of such theorems is usually in the form of the following sequence:
S1 ⇒ S2, ..., Sn−1 ⇒ Sn, Sn ⇒ S1 Example of such theorem from graph theory: The following conditions are equivalent: graph G is a tree graph G is acyclic and connected graph G is connected and has exactly |V | − 1 edges each edge in G is a bridge each pair of 2 vertices in G is connected by exactly 1 simple path adding any edge to G makes exactly 1 new cycle
Proving lists of equivalent statements
Discrete Mathematics Some theorems have the form: (c) Marcin The following statements are equivalent: S S S . Sydow 1, 2, ..., n
Proofs
Inference rules
Proofs
Set theory axioms graph G is a tree graph G is acyclic and connected graph G is connected and has exactly |V | − 1 edges each edge in G is a bridge each pair of 2 vertices in G is connected by exactly 1 simple path adding any edge to G makes exactly 1 new cycle
Proving lists of equivalent statements
Discrete Mathematics Some theorems have the form: (c) Marcin The following statements are equivalent: S S S . Sydow 1, 2, ..., n A typical proof of such theorems is usually in the form of the Proofs following sequence: Inference rules S1 ⇒ S2, ..., Sn−1 ⇒ Sn, Sn ⇒ S1 Proofs
Set theory Example of such theorem from graph theory: axioms The following conditions are equivalent: graph G is acyclic and connected graph G is connected and has exactly |V | − 1 edges each edge in G is a bridge each pair of 2 vertices in G is connected by exactly 1 simple path adding any edge to G makes exactly 1 new cycle
Proving lists of equivalent statements
Discrete Mathematics Some theorems have the form: (c) Marcin The following statements are equivalent: S S S . Sydow 1, 2, ..., n A typical proof of such theorems is usually in the form of the Proofs following sequence: Inference rules S1 ⇒ S2, ..., Sn−1 ⇒ Sn, Sn ⇒ S1 Proofs
Set theory Example of such theorem from graph theory: axioms The following conditions are equivalent: graph G is a tree graph G is connected and has exactly |V | − 1 edges each edge in G is a bridge each pair of 2 vertices in G is connected by exactly 1 simple path adding any edge to G makes exactly 1 new cycle
Proving lists of equivalent statements
Discrete Mathematics Some theorems have the form: (c) Marcin The following statements are equivalent: S S S . Sydow 1, 2, ..., n A typical proof of such theorems is usually in the form of the Proofs following sequence: Inference rules S1 ⇒ S2, ..., Sn−1 ⇒ Sn, Sn ⇒ S1 Proofs
Set theory Example of such theorem from graph theory: axioms The following conditions are equivalent: graph G is a tree graph G is acyclic and connected each edge in G is a bridge each pair of 2 vertices in G is connected by exactly 1 simple path adding any edge to G makes exactly 1 new cycle
Proving lists of equivalent statements
Discrete Mathematics Some theorems have the form: (c) Marcin The following statements are equivalent: S S S . Sydow 1, 2, ..., n A typical proof of such theorems is usually in the form of the Proofs following sequence: Inference rules S1 ⇒ S2, ..., Sn−1 ⇒ Sn, Sn ⇒ S1 Proofs
Set theory Example of such theorem from graph theory: axioms The following conditions are equivalent: graph G is a tree graph G is acyclic and connected graph G is connected and has exactly |V | − 1 edges each pair of 2 vertices in G is connected by exactly 1 simple path adding any edge to G makes exactly 1 new cycle
Proving lists of equivalent statements
Discrete Mathematics Some theorems have the form: (c) Marcin The following statements are equivalent: S S S . Sydow 1, 2, ..., n A typical proof of such theorems is usually in the form of the Proofs following sequence: Inference rules S1 ⇒ S2, ..., Sn−1 ⇒ Sn, Sn ⇒ S1 Proofs
Set theory Example of such theorem from graph theory: axioms The following conditions are equivalent: graph G is a tree graph G is acyclic and connected graph G is connected and has exactly |V | − 1 edges each edge in G is a bridge adding any edge to G makes exactly 1 new cycle
Proving lists of equivalent statements
Discrete Mathematics Some theorems have the form: (c) Marcin The following statements are equivalent: S S S . Sydow 1, 2, ..., n A typical proof of such theorems is usually in the form of the Proofs following sequence: Inference rules S1 ⇒ S2, ..., Sn−1 ⇒ Sn, Sn ⇒ S1 Proofs
Set theory Example of such theorem from graph theory: axioms The following conditions are equivalent: graph G is a tree graph G is acyclic and connected graph G is connected and has exactly |V | − 1 edges each edge in G is a bridge each pair of 2 vertices in G is connected by exactly 1 simple path Proving lists of equivalent statements
Discrete Mathematics Some theorems have the form: (c) Marcin The following statements are equivalent: S S S . Sydow 1, 2, ..., n A typical proof of such theorems is usually in the form of the Proofs following sequence: Inference rules S1 ⇒ S2, ..., Sn−1 ⇒ Sn, Sn ⇒ S1 Proofs
Set theory Example of such theorem from graph theory: axioms The following conditions are equivalent: graph G is a tree graph G is acyclic and connected graph G is connected and has exactly |V | − 1 edges each edge in G is a bridge each pair of 2 vertices in G is connected by exactly 1 simple path adding any edge to G makes exactly 1 new cycle To prove equality of two sets: A = B it is enough to prove two set inclusions: A ⊆ B and B ⊆ A, thus it is enough to prove the two implications of the above form.
Example: Proving set inclusion and set equality
Discrete Mathematics
(c) Marcin Sydow
Proofs To prove that some set is included in another set: A ⊆ B it is Inference enough to use the de nition of inclusion. Thus, it is enough to rules prove the implication: Proofs x A x B (where x is any element of the universe) Set theory ∀x ∈ ⇒ ∈ axioms Example: Proving set inclusion and set equality
Discrete Mathematics
(c) Marcin Sydow
Proofs To prove that some set is included in another set: A ⊆ B it is Inference enough to use the de nition of inclusion. Thus, it is enough to rules prove the implication: Proofs x A x B (where x is any element of the universe) Set theory ∀x ∈ ⇒ ∈ axioms To prove equality of two sets: A = B it is enough to prove two set inclusions: A ⊆ B and B ⊆ A, thus it is enough to prove the two implications of the above form. Russel's antinomy: Z = {x : x ∈/ x} Does Z belong to itself? x ∈ Z ⇔ x ∈/ x Z ∈ Z ⇔ Z ∈/ Z (a contradiction) Thus the existence of the set Z led to a contradiction.
Russels antinomy
Discrete Mathematics
(c) Marcin Sydow There does not exist the set of all sets.2 Proofs
Inference rules
Proofs
Set theory axioms
2we call the family of all the sets class x ∈ Z ⇔ x ∈/ x Z ∈ Z ⇔ Z ∈/ Z (a contradiction) Thus the existence of the set Z led to a contradiction.
Russels antinomy
Discrete Mathematics
(c) Marcin Sydow There does not exist the set of all sets.2 Proofs Russel's antinomy: Inference rules Z = {x : x ∈/ x} Proofs Does Z belong to itself? Set theory axioms
2we call the family of all the sets class Z ∈ Z ⇔ Z ∈/ Z (a contradiction) Thus the existence of the set Z led to a contradiction.
Russels antinomy
Discrete Mathematics
(c) Marcin Sydow There does not exist the set of all sets.2 Proofs Russel's antinomy: Inference rules Z = {x : x ∈/ x} Proofs Does Z belong to itself? Set theory axioms x ∈ Z ⇔ x ∈/ x
2we call the family of all the sets class Thus the existence of the set Z led to a contradiction.
Russels antinomy
Discrete Mathematics
(c) Marcin Sydow There does not exist the set of all sets.2 Proofs Russel's antinomy: Inference rules Z = {x : x ∈/ x} Proofs Does Z belong to itself? Set theory axioms x ∈ Z ⇔ x ∈/ x Z ∈ Z ⇔ Z ∈/ Z (a contradiction)
2we call the family of all the sets class Russels antinomy
Discrete Mathematics
(c) Marcin Sydow There does not exist the set of all sets.2 Proofs Russel's antinomy: Inference rules Z = {x : x ∈/ x} Proofs Does Z belong to itself? Set theory axioms x ∈ Z ⇔ x ∈/ x Z ∈ Z ⇔ Z ∈/ Z (a contradiction) Thus the existence of the set Z led to a contradiction.
2we call the family of all the sets class 1 Uniqueness Axiom (Axiom of extensionality): If the sets A and B have the same elements then A and B are identical. 2 Union Axiom: for arbitrary sets A and B there exists the set whose elements are all the elements of the set A and all the elements of the set B (without repetitions) and no other elements 3 Dierence Axiom: For arbitrary sets A and B there exists the set whose elements are those and only those elements of the set A which are not the elements of the set B. 4 Existence Axiom: There exists at least one set. (intersection, the existence of the empty set and all the basic set algebra theorems can be derived from the above axioms)
Basic Axioms of Set Algebra
Discrete Primitive concepts: Mathematics element of set (c) Marcin Sydow the relation of belonging to the set (x ∈ X )
Proofs
Inference rules
Proofs
Set theory axioms 2 Union Axiom: for arbitrary sets A and B there exists the set whose elements are all the elements of the set A and all the elements of the set B (without repetitions) and no other elements 3 Dierence Axiom: For arbitrary sets A and B there exists the set whose elements are those and only those elements of the set A which are not the elements of the set B. 4 Existence Axiom: There exists at least one set. (intersection, the existence of the empty set and all the basic set algebra theorems can be derived from the above axioms)
Basic Axioms of Set Algebra
Discrete Primitive concepts: Mathematics element of set (c) Marcin Sydow the relation of belonging to the set (x ∈ X )
Proofs 1 Uniqueness Axiom (Axiom of extensionality): If the sets A Inference rules and B have the same elements then A and B are identical.
Proofs
Set theory axioms 3 Dierence Axiom: For arbitrary sets A and B there exists the set whose elements are those and only those elements of the set A which are not the elements of the set B. 4 Existence Axiom: There exists at least one set. (intersection, the existence of the empty set and all the basic set algebra theorems can be derived from the above axioms)
Basic Axioms of Set Algebra
Discrete Primitive concepts: Mathematics element of set (c) Marcin Sydow the relation of belonging to the set (x ∈ X )
Proofs 1 Uniqueness Axiom (Axiom of extensionality): If the sets A Inference rules and B have the same elements then A and B are identical. Proofs 2 Union Axiom: for arbitrary sets A and B there exists the Set theory set whose elements are all the elements of the set A and all axioms the elements of the set B (without repetitions) and no other elements 4 Existence Axiom: There exists at least one set. (intersection, the existence of the empty set and all the basic set algebra theorems can be derived from the above axioms)
Basic Axioms of Set Algebra
Discrete Primitive concepts: Mathematics element of set (c) Marcin Sydow the relation of belonging to the set (x ∈ X )
Proofs 1 Uniqueness Axiom (Axiom of extensionality): If the sets A Inference rules and B have the same elements then A and B are identical. Proofs 2 Union Axiom: for arbitrary sets A and B there exists the Set theory set whose elements are all the elements of the set A and all axioms the elements of the set B (without repetitions) and no other elements 3 Dierence Axiom: For arbitrary sets A and B there exists the set whose elements are those and only those elements of the set A which are not the elements of the set B. Basic Axioms of Set Algebra
Discrete Primitive concepts: Mathematics element of set (c) Marcin Sydow the relation of belonging to the set (x ∈ X )
Proofs 1 Uniqueness Axiom (Axiom of extensionality): If the sets A Inference rules and B have the same elements then A and B are identical. Proofs 2 Union Axiom: for arbitrary sets A and B there exists the Set theory set whose elements are all the elements of the set A and all axioms the elements of the set B (without repetitions) and no other elements 3 Dierence Axiom: For arbitrary sets A and B there exists the set whose elements are those and only those elements of the set A which are not the elements of the set B. 4 Existence Axiom: There exists at least one set. (intersection, the existence of the empty set and all the basic set algebra theorems can be derived from the above axioms) 5: For every propositional function f(x) and for every set A there exists a set consisting of those and only those elements of the set A which satisfy f(x)
{x : f (x) ∧ x ∈ A}
6: for every set A there exists a set, denoted by 2A, whose elements are all the subsets of A 7 (Axiom of Choice): For every family R of non-empty disjoint sets there exists a set which has one and only one element in common with each of the sets of the family R.3 (now axioms 2,3 are super uous as they can be derived from the axioms 1 and 5-7)
More Set Theory Axioms
Discrete More advanced set theory needs additional axioms: Mathematics
(c) Marcin Sydow
Proofs
Inference rules
Proofs
Set theory axioms
3The axiom of choice is very strong and implies some non-intuitive theorems and is questioned by some mathematicians 6: for every set A there exists a set, denoted by 2A, whose elements are all the subsets of A 7 (Axiom of Choice): For every family R of non-empty disjoint sets there exists a set which has one and only one element in common with each of the sets of the family R.3 (now axioms 2,3 are super uous as they can be derived from the axioms 1 and 5-7)
More Set Theory Axioms
Discrete More advanced set theory needs additional axioms: Mathematics 5: For every propositional function f(x) and for every set A (c) Marcin Sydow there exists a set consisting of those and only those
Proofs elements of the set A which satisfy f(x)
Inference rules {x : f (x) ∧ x ∈ A} Proofs
Set theory axioms
3The axiom of choice is very strong and implies some non-intuitive theorems and is questioned by some mathematicians 7 (Axiom of Choice): For every family R of non-empty disjoint sets there exists a set which has one and only one element in common with each of the sets of the family R.3 (now axioms 2,3 are super uous as they can be derived from the axioms 1 and 5-7)
More Set Theory Axioms
Discrete More advanced set theory needs additional axioms: Mathematics 5: For every propositional function f(x) and for every set A (c) Marcin Sydow there exists a set consisting of those and only those
Proofs elements of the set A which satisfy f(x)
Inference rules {x : f (x) ∧ x ∈ A} Proofs
Set theory axioms 6: for every set A there exists a set, denoted by 2A, whose elements are all the subsets of A
3The axiom of choice is very strong and implies some non-intuitive theorems and is questioned by some mathematicians (now axioms 2,3 are super uous as they can be derived from the axioms 1 and 5-7)
More Set Theory Axioms
Discrete More advanced set theory needs additional axioms: Mathematics 5: For every propositional function f(x) and for every set A (c) Marcin Sydow there exists a set consisting of those and only those
Proofs elements of the set A which satisfy f(x)
Inference rules {x : f (x) ∧ x ∈ A} Proofs
Set theory axioms 6: for every set A there exists a set, denoted by 2A, whose elements are all the subsets of A 7 (Axiom of Choice): For every family R of non-empty disjoint sets there exists a set which has one and only one element in common with each of the sets of the family R.3
3The axiom of choice is very strong and implies some non-intuitive theorems and is questioned by some mathematicians More Set Theory Axioms
Discrete More advanced set theory needs additional axioms: Mathematics 5: For every propositional function f(x) and for every set A (c) Marcin Sydow there exists a set consisting of those and only those
Proofs elements of the set A which satisfy f(x)
Inference rules {x : f (x) ∧ x ∈ A} Proofs
Set theory axioms 6: for every set A there exists a set, denoted by 2A, whose elements are all the subsets of A 7 (Axiom of Choice): For every family R of non-empty disjoint sets there exists a set which has one and only one element in common with each of the sets of the family R.3 (now axioms 2,3 are super uous as they can be derived from the axioms 1 and 5-7) 3The axiom of choice is very strong and implies some non-intuitive theorems and is questioned by some mathematicians Similar axiomatic approach is possible (and takes place) in other mathematical theories (e.g. theory of natural numbers, geometry, etc.)
The role of axioms
Discrete Mathematics
(c) Marcin Sydow
Proofs The introduction of the axioms of the set theory (at the beg. of Inference rules the XX. century) eliminated the paradoxes and antinomies and Proofs cleaned the fundamentals of the theory. Set theory axioms The role of axioms
Discrete Mathematics
(c) Marcin Sydow
Proofs The introduction of the axioms of the set theory (at the beg. of Inference rules the XX. century) eliminated the paradoxes and antinomies and Proofs cleaned the fundamentals of the theory. Set theory axioms Similar axiomatic approach is possible (and takes place) in other mathematical theories (e.g. theory of natural numbers, geometry, etc.) Example tasks/questions/problems
Discrete Mathematics
(c) Marcin provide the de nition of formal proof Sydow describe at least 6 dierent inference rules Proofs describe the following proof schemas: direct proof, proof Inference rules by contraposition, reductio ad absurdum (proof by
Proofs contradiction) Set theory prove the following small theorems: axioms If an integer n is odd, then n2 is also odd If n is an integer and 3n + 2 is odd, then n is odd At least four of any 22 days must fall on the same day of the week in each case, try the following schemas (in the given order): direct proof, proof by contraposition, reductio ad absurdum (proof by contradiction). Discrete Mathematics
(c) Marcin Sydow
Proofs
Inference rules Proofs Thank you for your attention. Set theory axioms