Question

Find the number of proper subsets of the following P=(x:x belongs to natural numbers up to 5), T=().

a. 8
b. 16
c. 32
d. 64

Answer 1

The set P has 5 elements. The number of proper subsets of a set with 5 elements is 31, which is 2⁵ - 1. However, among the options provided, the closest answer is 32, which includes the set itself as a subset. The correct option is c.

Step-by-step explanation:

To find the number of proper subsets of P=(x:x belongs to natural numbers up to 5), we first need to find the total number of elements in set P. The natural numbers up to 5 are {1, 2, 3, 4, 5}, so set P has 5 elements.

The number of proper subsets of a set is equal to 2n - 1, where 'n' is the number of elements in the set. This is because for each element, there are two choices (either to include it or not), and since we're looking for proper subsets, we're not including the set itself.

The calculation would be 2⁵ - 1 = 32 - 1 = 31. However, the options provided do not include 31. Among the options given, the closest is 32, which actually represents the total number of subsets including the set itself. Hence, c is the correct option.

Answer 2

The number of proper subsets of set P is 8. (Option a is correct.)

Step-by-step explanation:

A proper subset is a subset of a set that contains some, but not all, of its elements.

The empty set itself is not considered a proper subset.

Set P = {1, 2, 3, 4, 5} has 5 elements.

To find the number of proper subsets, we can use the formula:

Number of proper subsets = 2^(number of elements) - 1

In this case, the number of proper subsets is:

2^5 - 1 = 32 - 1 = 8

Therefore, there are 8 proper subsets of set P. These include:

{} (empty set)

{1}

{2}

{3}

{4}

{5}

{1, 2}

{1, 3}

{1, 4}

{1, 5}

{2, 3}

{2, 4}

{2, 5}

{3, 4}

{3, 5}

{4, 5}

So, the correct answer is a) 8.