TRUTH TABLES

1. TRUTH TABLES Section 1.3

2. 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.

3. 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.

4. 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.

5. Truth Table for p vq • 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.

6. 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.

7. 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.

8. 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.

9. 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).

10. 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 pv q is given by ~(pv 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.