TRUTH TABLES

1. TRUTH TABLES

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 ||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.

6. 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.

7. Proof that p||q Ξ (p&&!q) || q

8. De Morgan’s Laws • The negation of the conjunction p && q is given by !(p && q) ≡ !p || !q. “Not p and q” is equivalent to “not p or not q.” • The negation of the disjunction p|| q is given by !(p|| q) ≡ !p && !q. “Not p or q” is equivalent to “not p and not q.”