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:

Open interval

Open Interval

Open interval

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

closed interval

Closed Interval

closed interval

  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 right

Open on Left and Closed on Right

open 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 right

Closed on Left and Open on Right

closed 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:

open on both sides

Open on Both Sides

open on both sides

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

closed on the right

Closed on the Right

closed on the right

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

open on the both

Open on the Both

open on the both

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

closed on left

Closed on Left

closed on left

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.

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