# Mathematics: Sets Relations Functions: Number of Subsets of a Set

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## Sub – Set

If and are any two sets such that each element of the set is an element of the set also, then is said to be a subset of .

Example:

If and

, then is a proper subset of ?

Solution:

• It is given that

• ,

• Clearly and

• We write and say that is a proper subset of .

## Number of Subsets of a Set

If A is a set with , then the number of subsets of and number of proper subsets of

### Subsets of Real Numbers

The set of natural numbers

The set of whole numbers

The set of Integers

The set of Rational numbers

The set of irrational numbers denoted by .

I.e. all real numbers that are not rational

These sets are subsets of the set of real numbers. Some of the obvious relations among these subsets are

### Intervals as Subsets of Real Numbers

If and , then we have the following types of intervals:

1. The set is called an open interval and is denoted by. On the number line it is shown as:

1. The set is called a closed interval and is denoted by. On the number line it is shown as:

1. The set is an interval, open on left and closed on right. It is denoted by . On the number line it is shown as: Open on Left and Closed on Rightopen on left and closed on right
1. The set is an interval, closed on left and open on right. It is denoted by . On the number line it is shown as: Closed on Left and Open on Rightclosed on left and open on right
1. The set is an interval, which is denoted by . It is open on both sides. On the number line it is shown as:

1. The set is an interval which is denoted by . It is closed on the right. On the number line it is shown as: