   # Prove each statement in 39-44. For every positive integer n , if A and B i , B 2 , B 3 . are any sets, 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 43ES
Textbook Problem
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## Prove each statement in 39-44.For every positive integer n, if A and B i , B 2 , B 3 . are any sets, 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

i=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:

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

Prove that i=1n(A×Bi)=A×(i=1nBi)

First prove that i=1n(A×Bi)A×(i=1nBi).

Suppose there is an element (x,y) in i=1n(A×Bi), by definition of general union,

(x,y)(A×Bi) for some i=1,2,.....,n

Now by definition of Cartesian product, it follows that xA and yBi for some i=1,2,.....,n.

Since yBi for some i=1,2,......,n by definition of general union yi=1n(Bi).

Since xA and yi=1n(Bi), by definition of Cartesian product, it follows that

(x,y)A× (i=1nB1).

Thus, xi=1n(A×Bi)xA×(i=1nBi)

Therefore, i=1n(A×Bi)A×(i=1nBi)

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