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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.

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.

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.

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 dont 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.

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.

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).

De Morgans 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 Morgans Laws again with Venn

Diagrams in Chapter 2.

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