Algebra and Trigonometry: Structure and Method, Book 2

2000th Edition

ISBN: 9780395977255

Author: MCDOUGAL LITTEL

Publisher: McDougal Littell

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Question

Chapter 2, Problem 18E

To determine

Construct a truth table and determine whether the statement is a tautology.

Expert Solution & Answer

Answer to Problem 18E

The given statement is a tautology.

Explanation of Solution

Given information :

The given statements is [(p→q)∧(q→r)]→(p→r) .

Calculation :

A compound statement that is true for all truth values of its component statement is called a tautology.

The truth table of [(p→q)∧(q→r)]→(p→r) is drawn below:

pqrp→qq→r(p→q)∧(q→r)p→r[(p→q)∧(q→r)]→(p→r)TTTTTTTTTTFTFFFTTFTFTFTTTFFFTFFTFTTTTTTTFTFTFFTTFFTTTTTTFFFTTTTT

Column 4 is constructed by using conditional with column 1 and 2.

Column 5 is constructed by using conditional with column 2 and 3.

Column 6 is constructed by using conjunction with column 4 and 5.

Column 7 is constructed by using conditional with column 1 and 3.

Column 8 is constructed by using conditional with column 6 and 7.

From the above truth table:

Since, the given statement is true for all truth values of its component statement, therefore it is a tautology.

Hence,

The given statement is a tautology.

Chapter 2, Problem 17E

Chapter 2, Problem 1MPS

Chapter 2 Solutions

Algebra and Trigonometry: Structure and Method, Book 2

Ch. 2.1 - Prob. 1OE Ch. 2.1 - Prob. 2OE Ch. 2.1 - Prob. 3OE Ch. 2.1 - Prob. 4OE Ch. 2.1 - Prob. 5OE Ch. 2.1 - Prob. 6OE Ch. 2.1 - Prob. 7OE Ch. 2.1 - Prob. 8OE Ch. 2.1 - Prob. 9OE Ch. 2.1 - Prob. 10OE

Ch. 2.1 - Prob. 11OE Ch. 2.1 - Prob. 12OE Ch. 2.1 - Prob. 13OE Ch. 2.1 - Prob. 14OE Ch. 2.1 - Prob. 15OE Ch. 2.1 - Prob. 16OE Ch. 2.1 - Prob. 17OE Ch. 2.1 - Prob. 1WE Ch. 2.1 - Prob. 2WE Ch. 2.1 - Prob. 3WE Ch. 2.1 - Prob. 4WE Ch. 2.1 - Prob. 5WE Ch. 2.1 - Prob. 6WE Ch. 2.1 - Prob. 7WE Ch. 2.1 - Prob. 8WE Ch. 2.1 - Prob. 9WE Ch. 2.1 - Prob. 10WE Ch. 2.1 - Prob. 11WE Ch. 2.1 - Prob. 12WE Ch. 2.1 - Prob. 13WE Ch. 2.1 - Prob. 14WE Ch. 2.1 - Prob. 15WE Ch. 2.1 - Prob. 16WE Ch. 2.1 - Prob. 17WE Ch. 2.1 - Prob. 18WE Ch. 2.1 - Prob. 19WE Ch. 2.1 - Prob. 20WE Ch. 2.1 - Prob. 21WE Ch. 2.1 - Prob. 22WE Ch. 2.1 - Prob. 23WE Ch. 2.1 - Prob. 24WE Ch. 2.1 - Prob. 25WE Ch. 2.1 - Prob. 26WE Ch. 2.1 - Prob. 27WE Ch. 2.1 - Prob. 28WE Ch. 2.1 - Prob. 29WE Ch. 2.1 - Prob. 30WE Ch. 2.1 - Prob. 31WE Ch. 2.1 - Prob. 32WE Ch. 2.1 - Prob. 33WE Ch. 2.1 - Prob. 1MRE Ch. 2.1 - Prob. 2MRE Ch. 2.1 - Prob. 3MRE Ch. 2.1 - Prob. 4MRE Ch. 2.1 - Prob. 5MRE Ch. 2.1 - Prob. 6MRE Ch. 2.1 - Prob. 7MRE Ch. 2.1 - Prob. 8MRE Ch. 2.1 - Prob. 9MRE Ch. 2.1 - Prob. 10MRE Ch. 2.1 - Prob. 11MRE Ch. 2.1 - Prob. 12MRE Ch. 2.1 - Prob. 1CE Ch. 2.1 - Prob. 2CE Ch. 2.1 - Prob. 3CE Ch. 2.1 - Prob. 4CE Ch. 2.2 - Prob. 1OE Ch. 2.2 - Prob. 2OE Ch. 2.2 - Prob. 3OE Ch. 2.2 - Prob. 4OE Ch. 2.2 - Prob. 5OE Ch. 2.2 - Prob. 6OE Ch. 2.2 - Prob. 7OE Ch. 2.2 - Prob. 8OE Ch. 2.2 - Prob. 9OE Ch. 2.2 - Prob. 10OE Ch. 2.2 - Prob. 11OE Ch. 2.2 - Prob. 12OE Ch. 2.2 - Prob. 13OE Ch. 2.2 - Prob. 14OE Ch. 2.2 - Prob. 1WE Ch. 2.2 - Prob. 2WE Ch. 2.2 - Prob. 3WE Ch. 2.2 - Prob. 4WE Ch. 2.2 - Prob. 5WE Ch. 2.2 - Prob. 6WE Ch. 2.2 - Prob. 7WE Ch. 2.2 - Prob. 8WE Ch. 2.2 - Prob. 9WE Ch. 2.2 - Prob. 10WE Ch. 2.2 - Prob. 11WE Ch. 2.2 - Prob. 12WE Ch. 2.2 - Prob. 13WE Ch. 2.2 - Prob. 14WE Ch. 2.2 - Prob. 15WE Ch. 2.2 - Prob. 16WE Ch. 2.2 - Prob. 17WE Ch. 2.2 - Prob. 18WE Ch. 2.2 - Prob. 19WE Ch. 2.2 - Prob. 20WE Ch. 2.2 - Prob. 21WE Ch. 2.2 - Prob. 22WE Ch. 2.2 - Prob. 23WE Ch. 2.2 - Prob. 24WE Ch. 2.2 - Prob. 25WE Ch. 2.2 - Prob. 26WE Ch. 2.2 - Prob. 27WE Ch. 2.2 - Prob. 28WE Ch. 2.2 - Prob. 29WE Ch. 2.2 - Prob. 30WE Ch. 2.2 - Prob. 31WE Ch. 2.2 - Prob. 32WE Ch. 2.2 - Prob. 33WE Ch. 2.2 - Prob. 34WE Ch. 2.2 - Prob. 35WE Ch. 2.2 - Prob. 36WE Ch. 2.3 - Prob. 1OE Ch. 2.3 - Prob. 2OE Ch. 2.3 - Prob. 3OE Ch. 2.3 - Prob. 4OE Ch. 2.3 - Prob. 5OE Ch. 2.3 - Prob. 6OE Ch. 2.3 - Prob. 7OE Ch. 2.3 - Prob. 8OE Ch. 2.3 - Prob. 9OE Ch. 2.3 - Prob. 10OE Ch. 2.3 - Prob. 11OE Ch. 2.3 - Prob. 12OE Ch. 2.3 - Prob. 13OE Ch. 2.3 - Prob. 14OE Ch. 2.3 - Prob. 15OE Ch. 2.3 - Prob. 16OE Ch. 2.3 - Prob. 1MRE Ch. 2.3 - Prob. 2MRE Ch. 2.3 - Prob. 3MRE Ch. 2.3 - Prob. 4MRE Ch. 2.3 - Prob. 5MRE Ch. 2.3 - Prob. 6MRE Ch. 2.3 - Prob. 7MRE Ch. 2.3 - Prob. 8MRE Ch. 2.3 - Prob. 9MRE Ch. 2.3 - Prob. 10MRE Ch. 2.3 - Prob. 1ST Ch. 2.3 - Prob. 2ST Ch. 2.3 - Prob. 3ST Ch. 2.3 - Prob. 4ST Ch. 2.3 - Prob. 5ST Ch. 2.3 - Prob. 6ST Ch. 2.3 - Prob. 7ST Ch. 2.4 - Prob. 1OE Ch. 2.4 - Prob. 2OE Ch. 2.4 - Prob. 3OE Ch. 2.4 - Prob. 4OE Ch. 2.4 - Prob. 5OE Ch. 2.4 - Prob. 6OE Ch. 2.4 - Prob. 7OE Ch. 2.4 - Prob. 8OE Ch. 2.4 - Prob. 9OE Ch. 2.4 - Prob. 10OE Ch. 2.4 - Prob. 11OE Ch. 2.4 - Prob. 1WE Ch. 2.4 - Prob. 2WE Ch. 2.4 - Prob. 3WE Ch. 2.4 - Prob. 4WE Ch. 2.4 - Prob. 5WE Ch. 2.4 - Prob. 6WE Ch. 2.4 - Prob. 7WE Ch. 2.4 - Prob. 8WE Ch. 2.4 - Prob. 9WE Ch. 2.4 - Prob. 10WE Ch. 2.4 - Prob. 11WE Ch. 2.4 - Prob. 12WE Ch. 2.4 - Prob. 13WE Ch. 2.4 - Prob. 14WE Ch. 2.4 - Prob. 15WE Ch. 2.4 - Prob. 16WE Ch. 2.4 - Prob. 17WE Ch. 2.4 - Prob. 18WE Ch. 2.4 - Prob. 19WE Ch. 2.4 - Prob. 20WE Ch. 2.4 - Prob. 21WE Ch. 2.4 - Prob. 22WE Ch. 2.4 - Prob. 23WE Ch. 2.4 - Prob. 24WE Ch. 2.4 - Prob. 25WE Ch. 2.4 - Prob. 26WE Ch. 2.4 - Prob. 27WE Ch. 2.4 - Prob. 28WE Ch. 2.4 - Prob. 29WE Ch. 2.4 - Prob. 30WE Ch. 2.4 - Prob. 31WE Ch. 2.4 - Prob. 32WE Ch. 2.4 - Prob. 33WE Ch. 2.4 - Prob. 34WE Ch. 2.4 - Prob. 1CE Ch. 2.4 - Prob. 2CE Ch. 2.5 - Prob. 1OE Ch. 2.5 - Prob. 2OE Ch. 2.5 - Prob. 3OE Ch. 2.5 - Prob. 4OE Ch. 2.5 - Prob. 5OE Ch. 2.5 - Prob. 6OE Ch. 2.5 - Prob. 7OE Ch. 2.5 - Prob. 8OE Ch. 2.5 - Prob. 9OE Ch. 2.5 - Prob. 10OE Ch. 2.5 - Prob. 11OE Ch. 2.5 - Prob. 12OE Ch. 2.5 - Prob. 13OE Ch. 2.5 - Prob. 14OE Ch. 2.5 - Prob. 15OE Ch. 2.5 - Prob. 16OE Ch. 2.5 - Prob. 17OE Ch. 2.5 - Prob. 18OE Ch. 2.5 - Prob. 19OE Ch. 2.5 - Prob. 20OE Ch. 2.5 - Prob. 21OE Ch. 2.5 - Prob. 22OE Ch. 2.5 - Prob. 23OE Ch. 2.5 - Prob. 24OE Ch. 2.5 - Prob. 1WE Ch. 2.5 - Prob. 2WE Ch. 2.5 - Prob. 3WE Ch. 2.5 - Prob. 4WE Ch. 2.5 - Prob. 5WE Ch. 2.5 - Prob. 6WE Ch. 2.5 - Prob. 7WE Ch. 2.5 - Prob. 8WE Ch. 2.5 - Prob. 9WE Ch. 2.5 - Prob. 10WE Ch. 2.5 - Prob. 11WE Ch. 2.5 - Prob. 12WE Ch. 2.5 - Prob. 13WE Ch. 2.5 - Prob. 14WE Ch. 2.5 - Prob. 15WE Ch. 2.5 - Prob. 16WE Ch. 2.5 - Prob. 17WE Ch. 2.5 - Prob. 18WE Ch. 2.5 - Prob. 19WE Ch. 2.5 - Prob. 20WE Ch. 2.5 - Prob. 21WE Ch. 2.5 - Prob. 22WE Ch. 2.5 - Prob. 23WE Ch. 2.5 - Prob. 24WE Ch. 2.5 - Prob. 25WE Ch. 2.5 - Prob. 26WE Ch. 2.5 - Prob. 27WE Ch. 2.5 - Prob. 28WE Ch. 2.5 - Prob. 29WE Ch. 2.5 - Prob. 30WE Ch. 2.5 - Prob. 31WE Ch. 2.5 - Prob. 32WE Ch. 2.5 - Prob. 33WE Ch. 2.5 - Prob. 1MRE Ch. 2.5 - Prob. 2MRE Ch. 2.5 - Prob. 3MRE Ch. 2.5 - Prob. 4MRE Ch. 2.5 - Prob. 5MRE Ch. 2.5 - Prob. 6MRE Ch. 2.5 - Prob. 7MRE Ch. 2.5 - Prob. 8MRE Ch. 2.5 - Prob. 9MRE Ch. 2.5 - Prob. 1ST Ch. 2.5 - Prob. 2ST Ch. 2.5 - Prob. 3ST Ch. 2.5 - Prob. 4ST Ch. 2.5 - Prob. 5ST Ch. 2.5 - Prob. 6ST Ch. 2.5 - Prob. 1E Ch. 2.5 - Prob. 2E Ch. 2.5 - Prob. 3E Ch. 2.5 - Prob. 4E Ch. 2.5 - Prob. 5E Ch. 2.5 - Prob. 6E Ch. 2.5 - Prob. 7E Ch. 2.5 - Prob. 8E Ch. 2.6 - Prob. 1OE Ch. 2.6 - Prob. 2OE Ch. 2.6 - Prob. 3OE Ch. 2.6 - Prob. 4OE Ch. 2.6 - Prob. 5OE Ch. 2.6 - Prob. 6OE Ch. 2.6 - Prob. 7OE Ch. 2.6 - Prob. 8OE Ch. 2.6 - Prob. 1WE Ch. 2.6 - Prob. 2WE Ch. 2.6 - Prob. 3WE Ch. 2.6 - Prob. 4WE Ch. 2.6 - Prob. 5WE Ch. 2.6 - Prob. 6WE Ch. 2.6 - Prob. 7WE Ch. 2.6 - Prob. 8WE Ch. 2.6 - Prob. 9WE Ch. 2.6 - Prob. 10WE Ch. 2.6 - Prob. 11WE Ch. 2.6 - Prob. 12WE Ch. 2.6 - Prob. 13WE Ch. 2.6 - Prob. 14WE Ch. 2.6 - Prob. 15WE Ch. 2.6 - Prob. 16WE Ch. 2.6 - Prob. 17WE Ch. 2.6 - Prob. 18WE Ch. 2.6 - Prob. 19WE Ch. 2.6 - Prob. 20WE Ch. 2.6 - Prob. 21WE Ch. 2.6 - Prob. 22WE Ch. 2.6 - Prob. 23WE Ch. 2.6 - Prob. 24WE Ch. 2.6 - Prob. 25WE Ch. 2.7 - Prob. 1OE Ch. 2.7 - Prob. 2OE Ch. 2.7 - Prob. 3OE Ch. 2.7 - Prob. 1WE Ch. 2.7 - Prob. 2WE Ch. 2.7 - Prob. 3WE Ch. 2.7 - Prob. 4WE Ch. 2.7 - Prob. 5WE Ch. 2.7 - Prob. 6WE Ch. 2.7 - Prob. 7WE Ch. 2.7 - Prob. 8WE Ch. 2.7 - Prob. 9WE Ch. 2.7 - Prob. 10WE Ch. 2.7 - Prob. 11WE Ch. 2.7 - Prob. 12WE Ch. 2.7 - Prob. 13WE Ch. 2.7 - Prob. 14WE Ch. 2.7 - Prob. 15WE Ch. 2.7 - Prob. 16WE Ch. 2.7 - Prob. 17WE Ch. 2.7 - Prob. 18WE Ch. 2.7 - Prob. 1MRE Ch. 2.7 - Prob. 2MRE Ch. 2.7 - Prob. 3MRE Ch. 2.7 - Prob. 4MRE Ch. 2.7 - Prob. 5MRE Ch. 2.7 - Prob. 6MRE Ch. 2.7 - Prob. 7MRE Ch. 2.7 - Prob. 8MRE Ch. 2.7 - Prob. 9MRE Ch. 2.7 - Prob. 1ST Ch. 2.7 - Prob. 2ST Ch. 2.7 - Prob. 3ST Ch. 2 - Prob. 1CR Ch. 2 - Prob. 2CR Ch. 2 - Prob. 3CR Ch. 2 - Prob. 4CR Ch. 2 - Prob. 5CR Ch. 2 - Prob. 6CR Ch. 2 - Prob. 7CR Ch. 2 - Prob. 8CR Ch. 2 - Prob. 9CR Ch. 2 - Prob. 10CR Ch. 2 - Prob. 11CR Ch. 2 - Prob. 12CR Ch. 2 - Prob. 13CR Ch. 2 - Prob. 1CT Ch. 2 - Prob. 2CT Ch. 2 - Prob. 3CT Ch. 2 - Prob. 4CT Ch. 2 - Prob. 5CT Ch. 2 - Prob. 6CT Ch. 2 - Prob. 7CT Ch. 2 - Prob. 8CT Ch. 2 - Prob. 9CT Ch. 2 - Prob. 10CT Ch. 2 - Prob. 11CT Ch. 2 - Prob. 12CT Ch. 2 - Prob. 13CT Ch. 2 - Prob. 14CT Ch. 2 - Prob. 15CT Ch. 2 - Prob. 16CT Ch. 2 - Prob. 1E Ch. 2 - Prob. 2E Ch. 2 - Prob. 3E Ch. 2 - Prob. 4E Ch. 2 - Prob. 5E Ch. 2 - Prob. 6E Ch. 2 - Prob. 7E Ch. 2 - Prob. 8E Ch. 2 - Prob. 9E Ch. 2 - Prob. 10E Ch. 2 - Prob. 11E Ch. 2 - Prob. 12E Ch. 2 - Prob. 13E Ch. 2 - Prob. 14E Ch. 2 - Prob. 15E Ch. 2 - Prob. 16E Ch. 2 - Prob. 17E Ch. 2 - Prob. 18E Ch. 2 - Prob. 1MPS Ch. 2 - Prob. 2MPS Ch. 2 - Prob. 3MPS Ch. 2 - Prob. 4MPS Ch. 2 - Prob. 5MPS Ch. 2 - Prob. 6MPS Ch. 2 - Prob. 7MPS Ch. 2 - Prob. 8MPS Ch. 2 - Prob. 9MPS Ch. 2 - Prob. 10MPS Ch. 2 - Prob. 11MPS Ch. 2 - Prob. 12MPS Ch. 2 - Prob. 13MPS Ch. 2 - Prob. 14MPS

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