TRUTH TABLES Section 1 3 Introduction The truth

TRUTH TABLES Section 1. 3

Introduction • The truth value of a statement is the classification as true or false which denoted by T or F. • A truth table is a listing of all possible combinations of the individual statements as true or false, along with the resulting truth value of the compound statements. • Truth tables are an aide in distinguishing valid and invalid arguments.

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

Truth Table for p ^ q • Recall that the conjunction is the joining of two statements with the word and. • The number of rows in this truth table will be 4. (Since p has 2 values, and q has 2 value. ) • 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. • The number of rows in this table will be 4, since we have two statements and they can take on the two values of true and false. • 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 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 T T F F T F T T

Number of Rows • If a compound statement consists of n individual statements, each represented by a different letter, the number of rows required in the truth table is 2 n.

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 ~q p ^ ~q p q ~(p q) T T F F T F T F F F T F

De Morgan’s Laws • The negation of the conjunction p ^ q is given by ~(p ^ q) ≡ ~p v ~q. “Not p and q” is equivalent to “not p or not q. ” • The negation of the disjunction p v q is given by ~(p v q) ≡ ~p ^ ~q. “Not p or q” is equivalent to “not p and not q. ” • We will look at De Morgan’s Laws again with Venn Diagrams in Chapter 2.