   # Let Z be the set of all integers and let A 0 = { n ∈ Z | n = 4 k , for some integer k } A 1 = { n ∈ Z | n = 4 k + 1 , for some integer k } A 2 = { n ∈ Z | n = 4 k + 2 , for some integer k } and A 3 = { n ∈ Z | n = 4 k + 3 , for some integer k } Is { A 0 , A 1 , A 2 , A 3 } a partition of Z? Explain your Is { A 0 , A 1 , A 2 , A 3 } a partition of Z ? Explain your answer. ### 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 6.1, Problem 30ES
Textbook Problem
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## Let Z be the set of all integers and let A 0 = { n ∈ Z | n = 4 k ,   for   some integer  k } A 1 = { n ∈ Z | n = 4 k + 1 ,   for   some integer  k } A 2 = { n ∈ Z | n = 4 k + 2 ,   for   some integer  k } and A 3 = { n ∈ Z | n = 4 k + 3 ,   for   some integer  k } Is { A 0 , A 1 , A 2 , A 3 } a partition of Z? Explain your Is { A 0 , A 1 , A 2 , A 3 } a partition of Z? Explain your answer.

To determine

### Explanation of Solution

Given information:

Let Z be the set of all integers and let

A0={nZ|n=4k, for some integer k}A1={nZ|n=4k+1, for some integer k}A2={nZ|n=4k+2, for some integer k} and A3={nZ|n=4k+3, for some integer k}

Concept used:

Z be the set of all integers.

Calculation:

Let Z be the set of all integers.

Consider the following sets.

A0={nZ|n=4k, for some integer k}={....8,4,0,4,8,....}A1={nZ|n=4k+1, for some integer k}={....7,3,1,5,....}A2={nZ|n=4k+2, for some integer k} and ={....6,2,2,6,10,....}A3={nZ|n=4k+3, for some integer k}={....5,1,3,7,11,....}

The objective is to check that the set {A0,A1,A2,A3} is partition of integer set Z.

Partition of the set:

Let the set {A1,A2,.....,An} be the subsets of A. The set {A1,A2,.....,An} Is said to be partition of A. if each the elements are mutually disjoint and A1A2.......An=A.

Check that whether the set {A0,A1,A2,A3} is mutually disjoint set or not.

By the definition of the sets A0 and A1, there are no common elements in both the sets

A0 and A1

Thus, A0A1=

By the definition of the sets A0 and A2, there are no common elements in both the sets

A0 and A2

Thus, A0A2=

By the definition of the sets A0 and A3, there are no common elements in both the sets

A0 and A3

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