TRUTH TABLES

T F Truth Table for p  q p q p  q Recall that conditional is a compound statement of the form “if p then 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. 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. p q p  q T F

T F Truth Table for p ^ q p q 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 F

T F Truth Table for p v q p q 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 p v q T F

T F p ~p 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

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 ~p ~(~p) T F

Negation of the Conditional Here we look at the negation of the conditional. Note that the 4th and 6th columns are identical. Hence p ^ ~q is equivalent to ~(p  q). p q ~q p ^ ~q p  q ~(p  q) T F