Discrete Mathematics (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 = fS1; S2; :::; Sng 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 = fP1; P2; :::; Pmg 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 = fP1; P2; :::; Pmg 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 = fS1; S2; :::; Sng of Proofs logical statements so that each statement Si is either: Set theory axioms Formal proof Discrete Mathematics (c) Marcin Sydow Let P = fP1; P2; :::; Pmg 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 = fS1; S2; :::; Sng 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 specic 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 specic 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 simplication 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 simplication 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 simplication 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 simplication 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 ) 8x P(x) 9x P(x) Existential instantiation ) P(c) for some element c P(c) for some element c Existential generalization ) 9x P(x) Inference rules for quantied predicates Discrete Rule of inference Name Mathematics (c) Marcin Sydow 8x P(x) Universal instantiation ) P(c) Proofs Inference rules Proofs Set theory axioms 9x P(x) Existential instantiation ) P(c) for some element c P(c) for some element c Existential generalization ) 9x P(x) Inference rules for quantied predicates Discrete Rule of inference Name Mathematics (c) Marcin Sydow 8x P(x) Universal instantiation ) P(c) Proofs Inference rules P c for an arbitrary c Universal generalization Proofs ( ) P x Set theory ) 8x ( ) axioms P(c) for some element c Existential generalization ) 9x P(x) Inference rules for quantied predicates Discrete Rule of inference Name Mathematics (c) Marcin Sydow 8x P(x) Universal instantiation ) P(c) Proofs Inference rules P c for an arbitrary c Universal generalization Proofs ( ) P x Set theory ) 8x ( ) axioms 9x P(x) Existential instantiation ) P(c) for some element c Inference rules for quantied predicates Discrete Rule of inference Name Mathematics (c) Marcin Sydow 8x P(x) Universal instantiation ) P(c) Proofs Inference rules P c for an arbitrary c Universal generalization Proofs ( ) P x Set theory ) 8x ( ) axioms 9x P(x) Existential instantiation ) P(c) for some element c P(c) for some element c Existential generalization ) 9x 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).