Write the contrapositive of the statement and use the contrapositive to determine whether the given statement is true or false. If a ≠ 3 . then | a | ≠ 3 .

Mathematical Excursions (MindTap C...

4th Edition
Richard N. Aufmann + 3 others
Publisher: Cengage Learning
ISBN: 9781305965584

Chapter
Section

Mathematical Excursions (MindTap C...

4th Edition
Richard N. Aufmann + 3 others
Publisher: Cengage Learning
ISBN: 9781305965584
Chapter 3.4, Problem 33ES
Textbook Problem
60 views

Write the contrapositive of the statement and use the contrapositive to determine whether the given statement is true or false.If a ≠ 3 . then | a | ≠ 3 .

To determine

To write:

The contrapositive of the statement and use the contrapositive to determine whether the given statement is true or false.

Explanation of Solution

Given information:

The statement If a3, then |a|3

Concept Involved:

A statement that can be written in if-then form is called Conditional Statement. The phrase immediately following the word if is the hypothesis. The phrase immediately following the word then is the conclusion.

A Conditional Statement can be represented in symbols as  pq, which is read " p implies q " or " if p, then q. "

Statements that have the same truth values are logically equivalent. They are either both TRUE or both FALSE.

If you change the hypothesis or conclusion of a conditional statement, you form related conditionals. The below chart shows the three related conditionals, converse, inverse and contrapositive, and how they are related to a conditional statement

 Logic Symbols Formed by Conditional p→q Using the given hypothesis and conclusion Converse q→p Exchanging the hypothesis and conclusion Inverse ~ p→~q Replacing the hypothesis with its negation and replacing the conclusion with its negation Contrapositive ~q→~p Negating the hypothesis, negating the conclusion, and switching them

A conditional statement is logically equivalent to its contrapositive.

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.

Calculation:

Statement: If a3, then |a|3

By looking at Statement, we can identify the hypothesis (

p ) and conclusion ( q ).

p:  a3

q:  |a|3

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