Number System and Arithmetic A subset is a set whose all elements are contained within another set. A subset is indicated by the symbol '⊆' and read as 'is a subset of' in set theory . In the figure below, every element of set A belongs to set B; A is called a subset of B. A subset may contain some or all elements of another set. Every set is a subset of itself, and the empty set ∅ is a subset of all sets. 1) For a set S = {x, y, z}, its the possible subsets are {}, {x}, {y}, {z}, {x, y}, {y, z}, {z, x} or {x, y, z} 2) If A = {odd numbers} and B = {1, 3, 5}, so B ⊆ A, and A is a superset of B. 3) If A = {apple, banana} and set B contains {all fruits}, then A is a subset of B. Subsets of Real Numbers Real numbers that can be expressed as decimal numbers fall into a variety of categories. From your daily existence, you are undoubtedly already familiar with fractions, decimals, and counting numbers. The following numbers are considered subset of real numbers: Subsets of Integers Integers are a set of numbers that include all the whole numbers (both positive and negative) along with zero. They do not include fractions, decimals, or numbers with a fractional component. The set of integers is typically denoted by the symbol Z. Example: To which subsets of the real numbers does -5 belong? -5 is a rational number and an integer. Power Set of a Set A set's power set consists of every subset as well as the original set and the empty set. P(A) stands for the power set of a given set A. For example, If A = {1, 2}, then P(A) = {{ }, {1}, {2}, {1, 2}}. Here we can clearly see that all the subsets of A are contained in the P(A) ,i.e., power set of A. Number of Subsets of a Set For any set A, number of subsets are given using the following formula Number of Subsets = 2 n Where n is number of elements in the set. As power set contain all the subsets of any set, thus for a set A which has 'n' elements then P(A) has 2 n elements. Example: How many elements of power set can be formed if there are four elements in a set? Number of elements of power set with four elements are 2 4 = 16. Types of Subsets in Maths There are two types of subsets that are: A proper subset only comprises a few members of the original set. Proper subset can never be equal to the original set. The number of elements of a proper subset is always less than the parent set. A proper subset is denoted by ⊂, We can express a proper subset for set A and set B as; A ⊂ B Example: Let set A = {1, 3, 5}, then proper subsets of A are {}, {1}, {3}, {5}, {1, 3} {3, 5} {1, 5}. Also, {1, 3, 5} is not a proper subset of A as the number of elements is not less than the number of elements of A. Number of Proper Subset: The number of proper subsets of a set with 'n' elements is 2 n - 1. Example: A set contain 3 elements, what will be the number of proper subsets? Number of proper subsets = 2 n - 1 N = 2 3 - 1 = 7 Improper Subset An "improper subset" of a set refers to the subset that is exactly the same as the original set itself. In other words, if you have a set A, the improper subset of A is A itself Example: What will be the improper subset of set A = {1, 3, 5}? Improper subset: {1,3,5} Number of improper subsets: For any set with nnn elements, there is exactly one improper subset — the set itself. Proper vs Improper Subsets The key differences between proper subsets and improper subsets are listed in the following table: Improper Subset It contains some of the elements of a set. It contains all the elements of a set. It will never equal to a give set. It is always equal to a given set. The number of proper subsets of a set with 'n' elements is 2 n - 1. For a collection of 'n' elements, the number of improper subsets is always 1. "⊂" symbol is used only for proper subsets. "⊆" symbol is used for improper subsets. Example: For a set P = {1,2} find proper and improper subset. Proper set is given by { }, {1} and {2} Improper set is given by {1,2} Subsets vs Supersets The key differences between both subsets and supersets are listed in the following table: Subset Formulas All the formulas related to subsets are give below. The number of subsets of a set with n elements is 2 n . This includes both proper and improper subsets. The number of proper subsets of a set with n elements is 2 n - 1. The number of improper subsets of any set is always 1. Related Articles Representation of Set Set Theory Formulas Solved Examples Problem 1: How many subsets in a set with 4 elements?

A set containing 4 elements will have 2 4 elements in it = 16. Problem 2: How many subsets in a set with 5 elements? A set containing 5 elements will have 2 5 elements in it = 32. Problem 3: Calculate the number of proper subsets for the given set A (A = 5, 6, 7, 8) . A proper subset of a set is a subset that is not equal to the original set and the set A = {5, 6, 7, 8} contains 4 elements. The number of proper subsets of a set with n elements is given by: 2ⁿ − 1 Substituting n = 4: 2⁴ − 1 = 16 − 1 = 15 Therefore, the number of proper subsets of set A is 15. Problem 4: Consider the following two sets: X = {a, b, c} and Y = {a, b, c}. Is set X a proper subset of set Y? For X to be a valid subset of Y, it must have fewer elements than Y and exclude at least one element from Y. All of the components in X are likewise present in Y, and both sets include the same elements. Therefore, X is not a proper subset of Y since it does not exclude any items and is equivalent to Y. Problem 5: A is a subset of B. If A = {x: x is an even natural number} and B = {y : y is a natural number}. The statement above is correct. A is subset of B because every elememt of A is also in B. A is proper subset of B because B contains elements that A does not( eg 1, 3, 5........) Practice Problems on Subsets in Maths Problem 1: Given a set A = {1, 2, 3, 4}, how many subsets does A have? Problem 2: List all the subsets of the set B = {a, b}. Problem 3: Given the set C = {x, y, z}, how many proper subsets does C have? Problem 4: Determine if D = {2, 3} is a subset of E = {1, 2, 3, 4}. Problem 5: Set P = {red, blue, green} and set Q = { }. Is Q a proper subset of P? Numbers in Maths 2 min read Arithmetic Operations 8 min read Fractions in Maths 3 min read Decimals in Maths 3 min read Exponents 4 min read Percentage 4 min read Variable in Maths 4 min read Polynomials 2 min read Coefficient 2 min read Algebraic Identities 14 min read Properties of Algebraic Operations 1 min read Lines and Angles 2 min read Geometric Shapes in Maths 2 min read Area and Perimeter 2 min read Surface Areas and Volumes 1 min read Points, Lines and Planes 11 min read Coordinate Axes and Coordinate Planes in 3D space 3 min read Trigonometric Ratios 3 min read Trigonometric Equations 7 min read Trigonometric Identities 7 min read Trigonometric Functions 2 min read Inverse Trigonometric Functions 8 min read Inverse Trigonometric Identities 3 min read Introduction to Differential Calculus 3 min read Limits in Calculus 7 min read Continuity of Functions 8 min read Differentiation 2 min read Differentiability of Functions 10 min read Integration 2 min read Basic Concepts of Probability 6 min read Bayes' Theorem 6 min read Probability Distribution 1 min read Descriptive Statistic for Data Science 5 min read Introduction to Inferential Statistics 5 min read Measures of Central Tendency in Statistics 8 min read Set Theory 2 min read NCERT Solutions for Class 8 to 12 7 min read RD Sharma Class 8 Solutions for Maths: Chapter Wise PDF 5 min read RD Sharma Class 9 Solutions 10 min read RD Sharma Class 10 Solutions 9 min read RD Sharma Class 11 Solutions for Maths 13 min read RD Sharma Class 12 Solutions for Maths 13 min read Placement 360 Course 2 min read Data Science 360 Course 2 min read GATE CS-IT & DA Courses 2 min read