TRUTH TABLES Truth Table for p q Recall
TRUTH TABLES
Truth Table for p q • Recall that conditional is a compound statement of the form “if p then q”. p q • Think of a conditional as a promise. If I don’t keep my promise, in other words q is false, then the conditional is false if the premise is true. T T F F T F T T • If I keep my promise, that is q is true, and the premise is true, then the conditional is true. • When the premise is false (i. e. p is false), then there was no promise. Hence by default the conditional is true.
Truth Table for p ^ q • Recall that the conjunction is the joining of two statements with the word “and”. • For p ^ q to be true, then both statements p, q, must be true. • If either statement or if both statements are false, then the conjunction is false. p q p^q T T F F T F T F F F
Truth Table for p v q • Recall that a disjunction is the joining of two statements with the word “or”. • For a disjunction to be true, at least one of the statements must be true. • A disjunction is only false, if both statements are false. p q pvq T T F F T F T T T F
Truth Table for ~p • Recall that the negation of a statement is the denial of the statement. • If the statement p is true, the negation of p, i. e. ~p is false. • If the statement p is false, then ~p is true. • Note that since the statement p could be true or false, we have 2 rows in the truth table. p ~p T F F T
Equivalent Expressions • Equivalent expressions are symbolic expressions that have identical truth values for each corresponding entry in a truth table. • Hence ~(~p) ≡ p. • The symbol ≡ means equivalent to. p T F ~p ~(~p) F T T F
Negation of the Conditional • Here we look at the negation of the conditional. • Note that the 4 th and 6 th columns are identical. • Hence p ^ ~q is equivalent to ~(p q). p q T T T F F ~q p ^ ~q p q ~(p q)
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