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 22.

Number
of Proper Subsets of the Set:

If
a set contains ‘n’ elements, then the number of proper subsets of the set is
2n – 1.

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

⇒ Number of proper subsets of A are 3 =
22 – 1 = 4 – 1

In
general, number of proper subsets of a given set = 2m –
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
23.

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

Number
of proper subsets are 7 = 8 – 1 = 23 – 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 = 22 = 4

Also,
the number of proper subsets = 22 – 1

                                                    = 4 – 1 = 3