Study proper Subset

In mathematics, a proper subset of a set S is denoted as S’ in which it is having some elements from a set S. The study of proper subset used a symbol `sub` for denoting a symbol. For example, a subset S’ of a set S is denoted as, S’ `sub`S. Now we are going to study about proper subset.

Explanation to study proper subset:

Consider a set {1, 2, 4, 6, 7}. From this set we can write the proper subsets are {1, 2, 4} , {1, 4, 6} , { 1, 2, 4, 7} etc. and {1, 2, 4, 6, 7} is not considered as subset.

Make sure that every element in proper subset must also in the given set. In generally a proper subset of set S is S’ `sub` S.

An empty set is a proper subset of all the sets which are not empty.

Example problem to study proper subset:

Example: 1

Which of the following is the proper subset of a set A = {7, 9, 10, 13}?

a) {1, 7, 9}

b) {7, 9 10, 13}

c) {9, 10, 13}

d) {9, 10, 13, 15}

Solution:

Given set A= {7, 9, 10, 13}

Option a: {1, 7, 9}

In this set the element 1 is not in the set A. So, it is not a proper subset of A.

Option b: {7, 9, 10, 13}

This set is same as the given set A. So, it is not a proper subset of A.

Option c: {9, 10, 13}

This set has the elements in the given set A and also it is not equal to A. So, it is the proper subset of A.

Answer: c

Example: 2

Which of the following is the proper subset of a set B = {m, n, p, q, r}?

a) {p, q}

b) {p,q, r, s}

c) {m, n, o, p}

d) {r, s}

Solution:

Option a: {p, q}

This set has the elements in the given set B and also it is not equal to B. So, it is the proper subset of B.

Answer: a

Practice problems to study proper subset:

Problem: 1

Which of the following is the proper subset of a set A = {m, n}

a) {m,n}

b) {1, 2}

c) {}

d) {m, n, o}

Answer: c

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Example: 2

Which of the following is the proper subset of a set A = {6, 7, 8, 9}

a) {6}

b) {8, 20}

c) {7, 14}

d) {m}

Answer: a