   # Prove each statement in 39-44. For every positive integer n , if A 1 , A 2 , A 3 , .... and B any are sets, then ∩ x = 1 ( n ) ( A i − B ) = ( ∩ x = 1 n A i ) − 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 42ES
Textbook Problem
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## Prove each statement in 39-44.For every positive integer n, if A 1 , A 2 , A 3 , .... and B any are sets, then ∩ x = 1 ( n ) ( A i − B ) = ( ∩ x = 1 n A i ) − B .

To determine

To Prove:

For all integers n1, if A1,A2,A3.... and B are any sets, then i=1n(AiB)=(i=1nAi)B.

### Explanation of Solution

Given information:

A1,A2,A3.... and B Are any sets.

Concept used:

:Union of sets:Intersection of sets

: subset of set

: Null set

Calculation:

Let B and A1,A2,A3,.........,An(n1) are any sets.

Prove that i=1n(AiB)=(i=1nAi)B

First prove that i=1n(AiB)(i=1nAi)B.

Suppose there is an element x in i=1n(AiB), by definition of intersection,

x(AiB) for all i=1,2,......,n

Since by definition of set difference for all i=1,2,......,n,xAi and xB.

Since xAi for all i=1,2,.....,n which implies that xi=1nAi.

Since xi=1nAi and xB, by definition of set difference x(i=1nAi)B.

Thus xi=1n(AiB)x(i=1nAi)B.

Therefore i=1n(AiB)(i=1nAi)B

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