2.2 Conditional Statements 1

Discrete Structures

Chapter 2: The Logic of Compound Statements

2.2 Conditional Statements

… hypothetical reasoning implies the subordination of the real to the realm of the possible… – Jean Piaget, 1896 – 1980

2.2 Conditional Statements 2

Logic

• The dean has announced that If the mathematics department gets an additional $40,000, then it will hire one new faculty member.

The above proposition is called a conditional proposition.

Why?

2.2 Conditional Statements 3

Conditional

• Definition– If p and q are the statement variables, the

conditional of q by p is “If p then q” or “p implies q” and is denoted p q. It is false when p is true and q is false. Otherwise, it is true.

–We call p the hypothesis (or antecedent) of the conditional.

– q is the conclusion (or consequent) of the conditional.

2.2 Conditional Statements 4

Example – pg. 49 #2

• Rewrite the statement in if-then form.– I am on time for work if I catch the 8:05 am bus.

2.1 Logical Forms and Equivalences 5

Conditional Truth Table

• The truth value for the conditional is summarized in the truth table on the right.

p q p q

T T

T F

F T

F F

2.2 Conditional Statements 6

Order of Operations

• According to the order of operations, – First – Second – Third – Fourth – Fifth

2.2 Conditional Statements 7

Example – pg. 49 #5

• Construct a truth table for the statement form p q q

p q p q p q p q q

T T

T F

F T

F F

conclusion hypothesis

2.2 Conditional Statements 8

Negation of a Conditional Statement

• By definition, pq is false iff its hypothesis, p, is true and its conclusion, q, is false. It follows that

(p q) p q

Proof:

2.2 Conditional Statements 9

Example – pg. 49 # 20 b

• Write the negations for each of the following statements.– If today is New Year’s Eve, then tomorrow is

January.

2.2 Conditional Statements 10

Contrapositive

• Definition– The contrapositive of a conditional statement of

the form “If p then q” is

If q then p

Symbolically, the contrapositive of

p q is q p

2.2 Conditional Statements 11

Example – pg. 49 # 22 b

• Write the contrapositive for the following statement.– If today is New Year’s Eve, then tomorrow is

January.

2.2 Conditional Statements 12

Converse & Inverse

• Definition– Suppose a conditional statement of the form

“If p then q” is given, • The converse is “If q then p.”• The inverse is “If p then not q.”

2.2 Conditional Statements 13

Example – pg. 49 # 23 b

• Write the converse and inverse for each statement:– If today is New Year’s Eve, then tomorrow is

January.

2.2 Conditional Statements 14

NOTE!

• A conditional statement and its converse are not logically equivalent.

• A conditional statement and its inverse are not logically equivalent.

• The converse and the inverse of a conditional statement are logically equivalent to each other.

2.2 Conditional Statements 15

Only If

• If p and q are statements, p only if q means “if not q then not p or “if p then q.”

2.2 Conditional Statements 16

Biconditional - iff

• Given the statement variables p and q, the biconditional of p and q is “p iff q” denoted pq. It is true if both p and q have the same truth values and is false if p and q have opposite truth values.

2.1 Logical Forms and Equivalences 17

Biconditional Truth Table

• The truth value for the biconditional is summarized in the truth table on the right.

p q p q

T T

T F

F T

F F

2.2 Conditional Statements 18

Example – pg. 50 # 32

• Rewrite the statements as a conjunction of two if-then statements.– This quadratic equation has two distinct real roots

if, and only if, its discriminate is greater than zero.

2.2 Conditional Statements 19

Necessary and Sufficient Conditions

• Definition– If r and s are statements:

• r is a sufficient condition for s means “if r then s.”

• r is a necessary condition for s means “if not r then not s.”

2.2 Conditional Statements 20

Example – pg. 50 # 41

• Rewrite the statement in if-then form.– Having two 45 angles is a sufficient condition for

this triangle to be a right triangle.