# 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:

The set is called an

__open interval__and is denoted by. On the number line it is shown as:

The set is called a

__closed interval__and is denoted by. On the number line it is shown as:

The set is an interval,

__open on left and closed on right__. It is denoted by . On the number line it is shown as:

The set is an interval,

__closed on left and open on right__. It is denoted by . On the number line it is shown as:

The set is an interval, which is denoted by . It is

__open on both sides__. On the number line it is shown as:

The set is an interval which is denoted by . It is

__closed on the right__. On the number line it is shown as:

The set is an interval which is denoted by . It is

__open on the both__sides. On the number line it is shown as:

The set is an interval which is denoted by . It is

__closed on left__. On the number line it is shown as:

First four intervals are called finite intervals and the number (which is always positive) is called the length of each of these four intervals and .

The last four intervals are called infinite intervals and length of these intervals is not defined.