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Problem 1 :

Let A  =  {1, 2, 3, 4, 5} and B  =  { 5, 3, 4, 2, 1}. Determine whether B is a proper subset of A.

Problem 2 :

Let A  =  {1, 2, 3, 4, 5} and B  =  {1, 2, 5}. Determine whether B is a proper subset of A.

Problem 3 :

Let A  =  {1, 2, 3, 4, 5} find the number of proper subsets of A.

Problem 4 :

Let A  =  {1, 2, 3 } find the power set of A.

Problem 5 :

Let A = {a, b, c, d, e}, find the cardinality of power set of A.

Answers

1. Answer :

A = {1, 2, 3, 4, 5}

B = { 5, 3, 4, 2, 1}

If B is the proper subset of A, every element of B must also be an element of A and also B must not be equal to A.

In the given sets A and B, every element of B is also an element of A. But B is equal A.

So, B is the subset of A, but not a proper subset.

2. Answer :

A = {1, 2, 3, 4, 5}

B = {1, 2, 5}

If B is the proper subset of A, every element of B must also be an element of A and also B must not be equal to A. 

In the given sets A and B, every element of B is also an element of A.

And also But B is not equal to A.

So, B is a proper subset of A. 

3. Answer :

A = {1, 2, 3, 4, 5}

Let the given set contains n number of elements.

Formula to find number of proper subsets :

= 2ⁿ - 1

The value of n for the given set A is 5.

Because the set A = {1, 2, 3, 4, 5} contains 5 elements. 

Number of proper subsets :

= 2⁵ - 1

= 32 - 1

= 31

4. Answer :

A = {1, 2, 3} 

We know that the power set is the set of all subsets.

Here, the given set A contains 3 elements.

Number of subsets = 2³ = 8.

Therefore,

P(A) = {{1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}, { }}

5. Answer :

A = {a, b, c, d, e}

Formula for cardinality of power set of A :

n[P(A)] = 2ⁿ

Here n stands for the number of elements contained by the given set A.

The given set A contains 5 elements. So, n = 5.

Then, we have

n[P(A)] = 2⁵

n[P(A)] = 32

So, the cardinality of the power set of A is 32.

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