# Subsets of a given Set

Number of Subsets of a given Set:

If a set contains ‘n’ elements, then the number of subsets of the set is 2$$^{2}$$.

Number of Proper Subsets of the Set:

If a set contains ‘n’ elements, then the number of proper subsets of the set is 2$$^{n}$$ - 1.

If A = {p, q} the proper subsets of A are [{ }, {p}, {q}]

⇒ Number of proper subsets of A are 3 = 2$$^{2}$$ - 1 = 4 - 1

In general, number of proper subsets of a given set = 2$$^{m}$$ - 1, where m is the number of elements.

For example:

1. If A {1, 3, 5}, then write all the possible subsets of A. Find their numbers.

Solution:

The subset of A containing no elements - {  }

The subset of A containing one element each - {1} {3} {5}

The subset of A containing two elements each - {1, 3} {1, 5} {3, 5}

The subset of A containing three elements - {1, 3, 5)

Therefore, all possible subsets of A are { }, {1}, {3}, {5}, {1, 3}, {3, 5}, {1, 3, 5}

Therefore, number of all possible subsets of A is 8 which is equal 2$$^{3}$$.

Proper subsets are = {  }, {1}, {3}, {5}, {1, 3}, {3, 5}

Number of proper subsets are 7 = 8 - 1 = 2$$^{3}$$ - 1

2. If the number of elements in a set is 2, find the number of subsets and proper subsets.

Solution:

Number of elements in a set = 2

Then, number of subsets = 2$$^{2}$$ = 4

Also, the number of proper subsets = 2$$^{2}$$ - 1

= 4 – 1 = 3

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

then the number of proper subsets = 2$$^{5}$$ - 1

= 32 - 1 = 31   {Take [2$$^{n}$$ - 1]}

and power set of A = 2$$^{5}$$ = 32 {Take [2$$^{n}$$]}

Set Theory

Sets

Objects Form a Set

Elements of a Set

Properties of Sets

Representation of a Set

Different Notations in Sets

Standard Sets of Numbers

Types of Sets

Pairs of Sets

Subset

Subsets of a Given Set

Operations on Sets

Union of Sets

Intersection of Sets

Difference of two Sets

Complement of a Set

Cardinal number of a set

Cardinal Properties of Sets

Venn Diagrams