   # Theorem: For all sets A and B , if A ⊆ B , then A ∪ B ⊆ B . ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193
Chapter 6.2, Problem 4ES
Textbook Problem
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## Theorem: For all sets A and B, if A ⊆ B , then A ∪ B ⊆ B . To determine

To fill:

The blanks in the proof of, for all sets AandB, if AB, then ABB.

### Explanation of Solution

Given:

 Statement Suppose A, B and C are any sets such that A⊆B starting point We must show that A∪B⊆B conclusion to be shown Let x be any element in A∪B . start of an element proof Then x is in A or x is in B. __(a)____ In case x is in A, then x is in B. __(b)____ In case x is in B, then x is in B. tautology (p→q) So in either case x is in B. proof by division into cases Then every element in A∪B is in B. since x could be any element of A∪B. Therefore, A∪B⊆B [as was to be shown] __(c)____

Concept used:

If an element xAB, then xAorxB.

AB, if and only if every element of A is also an element of B

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