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.

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 ^ qp ^ q

T T TT F FF T FF 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.

pp qq p p vv q q

T T TT F TF T TF F 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.

pp qq p p q q

T T TT F FF T TF F 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 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.

pp ~p~p ~(~p)~(~p)

T F TF 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).

pp qq ~q~q p ^ ~qp ^ ~q p p q q ~(p ~(p q) q)

T T F F T F

T F T T F T

F T F F T F

F F T 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.