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A proper subset is denoted by the symbol ‘⊂.’ If A is the proper subset of B, it is expressed as A ⊂ B . If D = {a, g, h}, its proper subsets are { }, {a}, {g}, {h}, {a, g}, {a, h}, and {g, h} whereas the set {a, g, h} is not a proper subset of A. Here is another example shown with a Venn diagram, which represents set A = {5, 8, 9} as a proper subset of set B = {1, 2, 3, 4, 5, 8, 9}: Again, a set of whole numbers, natural numbers, integers, rational numbers, or irrational numbers are all subsets of real numbers. Similarly, a set of real numbers is a subset of the complex numbers. Improper Subset An improper subset is a subset that contains every element of another set, making it exactly equal to that set. Thus, every set has only one improper subset, the set itself. An improper subset is denoted by the symbol ‘⊆.’ Set A is an improper subset of set B if A is equal to B and is written as A ⊆ B. If D = {a, g, h}, its improper subset is {a, g, h}. Here is another example with a Venn diagram, representing set A = {1, 2, 5, 7, 8, 9} as an improper subset of set B = {1, 2, 5, 7, 8, 9} Now, let us summarize the differences between a proper and an improper subset. Proper vs. Improper Subset Number of Subsets of a Set The number of subsets for any set with ‘n’ elements is given by the formula 2 n . It includes the empty set. Let D = {a, g, h} be a given set. It has 3 elements. The possible subsets of D are { }, {a}, {g}, {h}, {a, g}, {a, h}, {g, h}, and {a, g, h}. It has a total of 8 subsets. Here, instead of finding all the subsets, we will use the formula 2 n = 2 3 = 8, which gives us the number of subsets of set D Here are a few more examples, A set with 4 elements has 2 4 = 16 subsets A set with 6 elements has 2 6 = 64 subsets A set with 1 element has 2 1 = 2 subsets (the empty set ɸ and the original set) A set with 0 elements has 2 0 = 1 subset (the empty set ɸ) The number of proper subsets of any set with ‘n’ elements is given by the formula 2 n – 1 . For example, if D = {a, g, h}, it has 2 3 – 1 = 7 proper subsets. The number of improper subsets of any set with ‘n’ elements is always 1 . Therefore, the number of elements in a set does not affect the number of improper subsets. The power set is the collection of all the subsets of a given set, including the empty set and the original set. If A is a set with ‘n’ elements, its power set is denoted by P(A) with 2 n elements. If B = {1, 3, 5}, its P(B) = {{ }, {1}, {3}, {5}, {1, 3}, {1, 5}, {3, 5}, {1, 3, 5}} Here, we observe that B has n = 3 elements and P(B) has 2 n = 2 3 = 8 elements. Solved Examples Find the subsets, proper subsets, and their number of elements for the set A = {2, 5, 7, 12} Given A = {2, 5, 7, 12} Thus, the subsets are { }, {2}, {5}, {7}, {12}, {2, 5}, {2, 7}, {2, 12}, {5, 7}, {5, 12}, {7, 12}, {2, 5, 7}, {2, 5, 12}, {2, 7, 12}, {5, 7, 12}, and {2, 5, 7, 12} The proper subset is {2, 5, 7, 12} Now, set A has n = 4 elements. Thus, the subset has 2 n = 2 4 = 16 elements, and the proper subset is always 1. Determine whether A is a subset of B when A = {7, 9, 11}, and B = {1, 3, 5, 7, 9, 11}. Given A = {7, 9, 11} and B = {1, 3, 5, 7, 9, 11} From set A, 7 ∈ B 9 ∈ B 11 ∈ B Since all elements of set A are present in set B, A ⊆ B Thus, A is a subset of B. Last modified on July 12th, 2024 Join Our Newsletter © 2026 Mathmonks.com . All rights reserved. Reproduction in whole or in part without permission is prohibited.