What are Subsets in Math – Proper and Improper Subsets

Here you will learn what are subsets in math i.e. proper subsets and improper subsets with examples.

Let’s begin –

What are Subsets in Math ?

Definition : Let A and B be two sets. If every element of A is an element of B, then A is called a subset of B.

If A is a subset of B, we write A \(\subset\) B, which is read as “A is a subset of B” or “A is contained in B”.

Thus,  A \(\subseteq\) B  if  a \(\in\) A  \(\implies\)  a \(\in\) B.

The symbol “\(\implies\)” stands for “implies”.

If A is a subset of B, we say that B contains A or B is a super set of A and we write B \(\supset\) A.

If A is not a subset of B, we write A \(\nsubseteq\) B.

Improper Subset

Definition : Every set is a subset of itself and the empty set is subset of every set. These two subsets are called improper subsets.

Example : Let Set A = {1, 2, 3}. Write its improper subsets.

Solution : Since every set is a subset of itself. Therefore {1, 2, 3} is subset of A and empty set (\(\phi\)) is a subset of every set.

\(\implies\) {1, 2, 3} and \(\phi\) are improper subsets of the given subset A.

Proper Subset

Definition : A subset A of a set B is called a proper subset of B if A \(\ne\) B and we write A \(\subset\) B. In such a case we also say that B is a super set of A.

Thus, if A is a proper subset of B, then there exist an element x \(\in\) B such that x \(\notin\) A.

In the example given below set B is the proper subset of set A.

Example : Let Set A = {1, 2, 3}, Set B = {1} and Set C = {1, 4}.

Then {1} \(\subseteq\) {1, 2, 3} but {1, 4} \(\nsubseteq\) {1, 2, 3}

\(\implies\) Set B is the subset of A because every element of B is in Set A. But Set C is not the subset of A because element 4 is not in Set A.

Theorems on Subsets

(i) Every set is a subset of itself.

(ii) The empty set is a subset of every set.

(iii) The total number of subsets of a finite set containing n elements is \(2^n\).

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