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How many different elements does A" have when A has m

Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen ISBN: 9780073383095 37

Solution for problem 37E Chapter 2.1

Discrete Mathematics and Its Applications | 7th Edition

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Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Discrete Mathematics and Its Applications | 7th Edition

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Problem 37E

Problem 37E

How many different elements does A" have when A has m elements and n is a positive integer?

Step-by-Step Solution:

Solution:

Step 1:

 In this problem, we have to find that how many different elements does An have when A has m elements and n is a positive integer.

Step2:

 Let’s consider A have m element {a1,a2…….am}

n is a positive integer  n>0

An={a1,a2…….am}n

As we know about total number of element in a set is given by its power set P(A)=2m 

 If we have An   set with m elements then we can write as P(A)=(2m)n

So we can write =2m.n

Step 3:

Assume we have a set of two element {a1,a2} where ajA for all values of j.

So we can write as

A2=A.A={a1,a2}.{a1,a2}

Similarly for A3

A3=A.A.A={a1,a2,a3}.{a1,a2,a3}.{a1,a2,a3}

Therefore we can write as

A3=A.A.A={a1,a2,a3|aj A for all values of j}

Step 4: 

Now we will consider for the An

This can be written as

An=A.A.A.A….A( n times)

{a1,a2,a3…...an)|aj A for all values of j=1,2,3…..n}

Hence, we have 2m.n different elements where  An have m elements and n is a positive integer.

Step 2 of 4

Chapter 2.1, Problem 37E is Solved
Step 3 of 4

Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

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