Statement of Proposition CSCI 1900 Discrete Structures • Statement of proposition – a declarative sentence that is either true or false, but not both Logical Operations • Examples: Reading: Kolman, Section 2.1 – The earth is round: statement that is true 2+3=5: statement that is true – Do you speak English? This is a question, not a statement CSCI 1900 – Discrete Structures Logical Operations – Page 1 CSCI 1900 – Discrete Structures Logical Operations – Page 2 More Examples of Statements of Logical Connectives and Proposition Compound Statements • 3-x=5: is a declarative sentence, but not a • x, y, z, … denote variables that can statement since it is true or false depending on the value of x represent real numbers • Take two aspirins: is a command, not a • p, q, r,… denote prepositional variables statement that can be replaced by statements. • The temperature on the surface of the planet Venus is 800oF: is a declarative statement of – p: The sun is shining today whose truth is unknown to us – q: It is cold • The sun will come out tomorrow: a statement that is either true or false, but not both, although we will have to wait until tomorrow to determine the answer CSCI 1900 – Discrete Structures Logical Operations – Page 3 CSCI 1900 – Discrete Structures Logical Operations – Page 4 Negation Examples of Negation • If p is a statement, the negation of p is the • If p: 2+3 >1 then If ~p: 2+3 <1 statement not p • If q: It is cold then ~q: It is not the case • Denoted ~p that it is cold, i.e., It is not cold. • If p is true, ~p is false • If p is false, ~p is true • ~p is not actually connective, i.e., it doesn’t join two of anything • not is a unary operation for the collection of statements and ~p is a statement if p is CSCI 1900 – Discrete Structures Logical Operations – Page 5 CSCI 1900 – Discrete Structures Logical Operations – Page 6 1 Conjunction Disjunction • If p and q are statements, then the • If p and q are statements, then the conjunction of p and q is the compound disjunction of p and q is the compound statement “p and q” statement “p or q” • Denoted p∧q • Denoted p∨q •p∧q is true only if both p and q are true •p∨q is true if either p or q are true • Example: • Example: – p: ETSU parking permits are expensive – p: I am a male – q: ETSU has plenty of parking – q: I am under 40 years old –p∧q = ? –p∨q = ? CSCI 1900 – Discrete Structures Logical Operations – Page 7 CSCI 1900 – Discrete Structures Logical Operations – Page 8 Exclusive Disjunction Inclusive Disjunction • If p and q are statements, then the • If p and q are statements, then the exclusive disjunction is the compound inclusive disjunction is the compound statement, “either p or q may be true, but statement, “either p or q may be true or both are not true at the same time.” they may both be true at the same time.” • Example: • Example: – p: It is daytime – p: It is cold – q: It is night time – q: It is night time –p∨q (in the exclusive sense) = ? –p∨q (in the inclusive sense) = ? CSCI 1900 – Discrete Structures Logical Operations – Page 9 CSCI 1900 – Discrete Structures Logical Operations – Page 10 Exclusive versus Inclusive Compound Statements • Depending on the circumstances, some •A compound statement is a statement made disjunctions are inclusive and some of exclusive. from other statements • Examples of Inclusive • For n individual propositions, there are 2n – “I have a dog” or “I have a cat” possible combinations of truth values – “It is warm outside” or “It is raining” • A truth table contains 2n rows identifying the • Examples of Exclusive truth values for the statement represented by the – Today is either Tuesday or it is Thursday table. – Pat is either male or female • Use parenthesis to denote order of precedence • ∧ has precedence over ∨ CSCI 1900 – Discrete Structures Logical Operations – Page 11 CSCI 1900 – Discrete Structures Logical Operations – Page 12 2 Truth Tables are Important Tools Compound Statement Example for this Material! (p ∧ q) ∨ (~p) p q p∧q p q p∨q p q p ∧ q ~p (p ∧ q) ∨ (~p) T T T T T T T T T F T T F F T F T F T F F T T T F F F F F F F F F F F T F T T F F F T T CSCI 1900 – Discrete Structures Logical Operations – Page 13 CSCI 1900 – Discrete Structures Logical Operations – Page 14 Quantifiers Computer Science Functions • Back in Section 1.1, a set was defined • if P(x), then execute certain steps {x | P(x)} • while Q(x), do specified actions • For an element t to be a member of the set, P(t) must evaluate to “true” • P(x) is called a predicate or a propositional function CSCI 1900 – Discrete Structures Logical Operations – Page 15 CSCI 1900 – Discrete Structures Logical Operations – Page 16 Universal quantification of a Examples: predicate P(x) • Universal quantification of predicate P(x) = • P(x): -(-x) = x For all values of x, P(x) is true – This predicate makes sense for all real • Denoted ∀x P(x) numbers x. • The symbol ∀ is called the universal – The universal quantification of P(x), ∀x P(x), is a true statement, because for all real quantifier numbers, -(-x) = x • The order in which multiple quantifications • Q(x): x+1<4 are considered does not affect the truth – ∀x Q(x) is a false statement, because, for value (e.g., ∀x ∀y P(x,y) ≡ ∀y ∀x P(x,y) ) example, Q(5) is not true CSCI 1900 – Discrete Structures Logical Operations – Page 17 CSCI 1900 – Discrete Structures Logical Operations – Page 18 3 Applying both universal and Existential quantification of a existential quantification predicate P(x) • Order of application does matter • Existential quantification of a predicate P(x) is • Example: Let A and B be n x n matrices the statement “There exists a value of x for • The statement ∀A ∃BA+ B = I which P(x) is true.” n • Reads “for every A there is a B such that A + B = • Denoted ∃x P(x) In” • Existential quantification may be applied to • Prove by coming up for equations for bii and bij (j≠i) several variables in a predicate • Now reverse the order: ∃B ∀AA+ B = In • Reads “there exists a B such that for all AA+ B = • The order in which multiple quantifications are In” considered does not affect the truth value • THIS IS FALSE! CSCI 1900 – Discrete Structures Logical Operations – Page 19 CSCI 1900 – Discrete Structures Logical Operations – Page 20 Okay, what exactly did the Assigning Quantification to previous slide say? Proposition • Assume a statement is made that “for all x, • Let p: ∀x P(x) P(x) is true.” • The negation of p is false when p is true – If we can find one case that is not true, then the and true when p is false statement is false. • For p to be false, there must be at least – If we cannot find one case that is not true, then the statement is true. one value of x for which P(x) is false. •Example: ∀ positive integers, n, • Thus, p is false if ∃x ~P(x) is true. P(n) = n2 + 41n + 41 is a prime number. • If ∃x ~P(x) is false, then for every x, ~P(x) – This is false because ∃ an integer resulting in a is false; that is ∀x P(x) is true. non-prime value, i.e., ∃n such that P(n) is false. CSCI 1900 – Discrete Structures Logical Operations – Page 21 CSCI 1900 – Discrete Structures Logical Operations – Page 22 4.