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 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.

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 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.

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 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.