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

To determine

To Prove:

For all integers n1 if A and B1,B2,B3,...... are any sets, then

n=1n(A×Bi)=A×(i=1nBi)

### Explanation of Solution

Given information:

Let A and B1,B2,B3,..........,Bn(n1) are any sets.

Concept used:

:Union of sets:Intersection of sets

: subset of set

: Null set

Calculation:

This prove n=1n(A×Bi)=A×(i=1nBi) is done in following ways.

Suppose A and B1,B2,B3,..........,Bn are any sets.

Proof that n=1n(A×Bi)A×(i=1nBi).

Suppose (x,y) is any element in n=1n(A×Bi).

By definition of intersection

(x,y)A×Bi For Cartesian product

xA(2) and yB1

1. for some
2. Implies that yi=1nBi.

Thus xA and yi=1nBi and so by definition of Cartesian product

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