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\(^{n}\)]}

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