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.

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

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. 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 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. 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 p v q T F

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

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 2n.

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

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.