TRUTH TABLES

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.

pp !p!p

T FF 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.

pp qq p && p && qq

T T TT F FF T FF F F

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.

pp qq p p |||| q q

T T TT F TF T TF F 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.

pp !p!p !(!p)!(!p)

T F TF T F

Proof that p||q Ξ (p&&!q) || q

pp qq !q!q p && !qp && !q p || qp || q (p&&!q) || q(p&&!q) || q

T T F F T T

T F T T T T

F T F F T T

F F T F F F

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.”