Sets form the foundation of modern mathematics. A set is a well-defined collection of distinct objects. The word "well-defined" means that given any object, we can definitely say whether it belongs to the collection or not. Sets are used to organise and represent collections of numbers, shapes, people, or any well-defined objects, and they appear in virtually every branch of mathematics.
This chapter introduces the language of sets: how to describe a set using the roster or tabular form and the set-builder form, the different types of sets such as empty, finite, infinite, equal, and equivalent sets, and the concept of subsets including the universal set and power set. It then develops operations on sets such as union, intersection, and difference, and establishes the fundamental laws of algebra of sets including De Morgan's laws.
Sets also provide the notation and machinery needed for later chapters on relations, functions, probability, and reasoning. The idea of membership, the symbols of union and intersection, and the complement operation are used repeatedly throughout Class 11 and Class 12 mathematics, so mastering this chapter pays dividends across the entire syllabus.
A set is a well-defined collection of objects. Each object in a set is called an element or member of the set. Sets are usually denoted by capital letters A, B, C, and elements by lower-case letters a, b, c.
The membership of an element x in a set A is written as x in A, read as "x belongs to A". If x is not an element of A, we write x not in A.
In this form, all elements are listed within curly braces, separated by commas. Examples:
A = {1, 2, 3, 4, 5} B = {a, e, i, o, u}
In the roster form, the order of elements does not matter, and repeated elements are written only once. For example, the set of letters in the word "MATHEMATICS" is {M, A, T, H, E, I, C, S}.
In this form, we write a property that the elements must satisfy. The set of all x such that x has property P is written as:
{x : P(x)}
For example, the set of all natural numbers less than 6 is written as {x : x in N and x < 6}, which equals {1, 2, 3, 4, 5}.
A set that contains no element is called the empty set or null set or void set, denoted by phi or {}. For example, the set of natural numbers between 3 and 4 is the empty set, since there is no natural number strictly between 3 and 4.
A set which has only one element is called a singleton set. For example, {0} and {5} are singleton sets.
A set is finite if its elements can be counted and the process of counting terminates. A set that is not finite is called infinite. The number of elements in a finite set A is denoted by n(A) and is called its cardinal number.
For example, the set {1, 2, 3} has cardinal number 3. The set of all natural numbers N is infinite.
Two sets A and B are said to be equal if every element of A is in B and every element of B is in A. This is written as A = B. Equal sets must have the same elements and hence the same cardinal number, but two sets with the same number of elements need not be equal.
Example: A = {1, 3, 5} and B = {3, 5, 1} are equal. A = {1, 2} and B = {2, 1} are equal, but A = {1, 2} and C = {1, 2, 3} are not equal.
Two finite sets A and B are equivalent if they have the same number of elements, that is, n(A) = n(B). Equivalent sets need not be equal. For example, A = {a, b, c} and B = {1, 2, 3} are equivalent but not equal.
If every element of set A is also an element of set B, then A is called a subset of B, written as A subset B. If A is not a subset of B, we write A not subset B. Every set is a subset of itself, and the empty set is a subset of every set.
A subset A of B such that A is not equal to B is called a proper subset of B. If A is a proper subset of B, then every element of A is in B and B has at least one element not in A.
A set with n elements has exactly 2^n subsets. For example, the set {a, b} has 4 subsets: {}, {a}, {b}, {a, b}. In general, if n(A) = n, then the number of subsets of A is 2^n and the number of proper subsets is 2^n - 1.
If A is a subset of B, then B is called a superset of A. Real number intervals are important subsets of R:
The universal set is the superset of which all sets under consideration are subsets. It is usually denoted by U. For example, when discussing sets of numbers, U may be taken as the set of real numbers R.
The power set of a set A is the collection of all subsets of A. It is denoted by P(A). If A has n elements, then P(A) has 2^n elements.
Example: If A = {1, 2}, then P(A) = {phi, {1}, {2}, {1, 2}}. Note that phi and A are both elements of P(A).
The union of two sets A and B is the set of all elements which belong to A or to B or to both. It is denoted by A union B, read as "A union B".
A union B = {x : x in A or x in B}
The intersection of two sets A and B is the set of all elements which belong to both A and B. It is denoted by A intersection B.
A intersection B = {x : x in A and x in B}
If A intersection B = phi, the sets are called disjoint sets.
The difference of A and B, written A - B, is the set of all elements which belong to A but not to B.
A - B = {x : x in A and x not in B}
Similarly, B - A = {x : x in B and x not in A}. In general, A - B is not equal to B - A.
The complement of set A with respect to the universal set U is the set of all elements of U which are not in A. It is denoted by A' or A^c.
A' = {x : x in U and x not in A}
Some properties of complement:
The last two are called De Morgan's laws.
If A and B are finite sets, then the number of elements in A union B is given by:
n(A union B) = n(A) + n(B) - n(A intersection B)
If A and B are disjoint, then n(A union B) = n(A) + n(B). For three sets:
n(A union B union C) = n(A) + n(B) + n(C) - n(A intersection B) - n(B intersection C) - n(A intersection C) + n(A intersection B intersection C)
Example: In a group of 100 students, 60 like tea and 50 like coffee. If 30 like both, the number who like tea or coffee is 60 + 50 - 30 = 80.
The operations of union, intersection, and complement satisfy several laws:
A union A = A, A intersection A = A
A union phi = A, A intersection U = A, A union U = U, A intersection phi = phi
A union B = B union A, A intersection B = B intersection A
(A union B) union C = A union (B union C) (A intersection B) intersection C = A intersection (B intersection C)
A union (B intersection C) = (A union B) intersection (A union C) A intersection (B union C) = (A intersection B) union (A intersection C)
(A union B)' = A' intersection B' (A intersection B)' = A' union B'
These laws are dual in the sense that if union and intersection are interchanged, the law remains valid. This principle is known as the principle of duality.
| Concept | Roster Form | Set-Builder Form | Cardinal Number |
|---|---|---|---|
| Natural numbers less than 6 | {1, 2, 3, 4, 5} | {x : x in N, x < 6} | 5 |
| Even integers | {..., -4, -2, 0, 2, 4, ...} | {x : x = 2k, k in Z} | Infinite |
| Solution set of x^2 = 9 | {-3, 3} | {x : x^2 = 9} | 2 |
| Empty set | {} | {x : x not equal to x} | 0 |
| Operation | Definition | Example (U = {1,2,3,4,5}, A = {1,2,3}, B = {3,4}) |
|---|---|---|
| Union A union B | Elements in A or B | {1, 2, 3, 4} |
| Intersection A intersection B | Elements in A and B | {3} |
| Difference A - B | Elements in A but not B | {1, 2} |
| Complement A' | Elements in U but not A | {4, 5} |
| Power set P(A) | All subsets of A | {phi, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}} |
Sets provide the precise language in which nearly all of mathematics is expressed. The concepts of membership, subsets, power sets, and the operations of union, intersection, and complement are indispensable tools for defining relations, functions, probability spaces, and logical statements. The laws of algebra of sets, particularly De Morgan's laws and the principle of duality, reveal deep symmetries that reappear in logic, boolean algebra, and computer science. A solid command of set notation and counting formulas such as the inclusion-exclusion principle will make all subsequent chapters, especially relations, functions, and probability, considerably easier to master.
Example 1: Finding the Number of Students. In a class survey, 40 students play cricket, 30 play football, and 12 play both games. To find how many students play at least one of the two games, we apply the inclusion-exclusion formula. Let A be the set of cricket players and B the set of football players. Then n(A) = 40, n(B) = 30, and n(A intersection B) = 12. Substituting into n(A union B) = n(A) + n(B) - n(A intersection B), we get 40 + 30 - 12 = 58. Therefore 58 students play cricket or football or both. The subtraction of 12 is crucial, because the 12 students who play both games are counted twice in the sum 40 + 30, once inside each set. This is the essence of the inclusion-exclusion principle, and it explains why we must always subtract the intersection whenever the sets overlap.
Example 2: Number of Proper Subsets. If a set has 6 elements, then the total number of subsets is 2^6 = 64, and the number of proper subsets is 2^6 - 1 = 63. The single subset that is not proper is the set itself, which is why we subtract exactly one. The empty set is counted as a proper subset because it is not equal to the original set. This reasoning is tested frequently in short-answer questions, so the formulas 2^n for subsets and 2^n - 1 for proper subsets must be applied with confidence and speed.
Example 3: Verifying De Morgan's Laws. Let U = {1, 2, 3, 4, 5}, A = {1, 2} and B = {2, 3}. First, A union B = {1, 2, 3}, so the complement (A union B)' = {4, 5}. On the other side, A' = {3, 4, 5} and B' = {1, 4, 5}, so the intersection A' intersection B' = {4, 5}. Both sides produce the same set {4, 5}, confirming that the complement of a union is the intersection of the complements. Repeating such verification with small sets builds a deep and intuitive grasp of De Morgan's laws, which is far more reliable than memorising the formulas alone, and it also prepares the student for the parallel laws used later in logic and Boolean algebra.