SET
The concept of set is fundamental in all branches of mathematics. A set is a well defined collection or ensemble of distinct objects. Well defined collection means that there exists a rule with the help of which it is possible to tell whether a given object belongs or does not belong to given collection. Generally sets are denoted by capital letters A, B, C, X, Y, Z etc.
Representation of set
The set may be described by listing the objects/elements belonging to it, each element being separated by comma. This method is called tabular form of the set. The set also may be described by stating the property which its elements must satisfy. A = {x: P(x)} means A is the set consisting of the elements x such that x satisfies the property P(x). This method is called Set Builder form of the set.
For example
A = {1, 2, 3, 4, 5} Tabular form; A = {x | x ≤ 5, x ∈ N} Set Builder form
Union of Sets
Union of two or more sets is the set of all elements that belong to any of these sets. The symbol used for union of sets is '∪' i.e. A ∪ B = Union of set A and set B = {x: x ∈ in A or x ∈ in B (or both)}
e.g. A = {1, 2, 3, 4} and B = {2, 4, 5, 6} and C = {1, 2, 6, 8}, then A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 8}
Intersection of Sets
It is the set of all the elements, which are common to all the sets. The symbol used for intersection of sets is ‘∩’ i.e. A ∩ B = {x: x ∈ in A and x ∈ B}
Difference of two sets
The difference of set A to B denoted as A − B is the set of those elements that are in the set A but not in the set B i.e. A − B = {x: x ∈ A and x ∉ B}
Similarly B − A = {x: x ∈ B and x ∉ A} In general A − B ≠ B − A
Symmetric Difference of Two Sets
For two sets A and B, symmetric difference of A and B is given by (A − B) ∪ (B − A) and is denoted by A Δ B.
Subset of a Set
A set A is said to be a subset of the set B if each element of the set A is also the element of the set B. The symbol used is ‘⊆’ i.e. A ⊆ B ⇔ (x ∈ A ⇒ x ∈ B).
Each set is a subset of its own set. Also a void set is a subset of any set. If there is at least one element in B which does not belong to the set A, then A is a proper subset of set B and is denoted as A ⊂ B.
Total number of subsets of a finite set containing n elements is 2ⁿ.
Cardinality of set
If A is a set, then number of elements in set A is known as cardinality of set A denoted by n(A).
e.g., A = {1, 2, 3}, n(A) = 3
Some More Results Regarding the Order of Finite Sets
Let A, B and C be finite sets and U be the finite universal set, then
(i) n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
(ii) n(A – B) = n(A) – n(A ∩ B)
(iii) n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C)
Ex.:
If A and B be two sets containing 3 and 6 elements respectively, then minimum number of elements in A ∪ B is
(A) 3
(B) 4
(C) 6
(D) 12
Solution:
(C). We have, n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
This shows that n(A ∪ B) is minimum according as n(A ∩ B) is maximum. When n(A ∩ B) is maximum;
This is possible only when A ⊆ B; In this case n(A ∩ B) = 3
∴ n(A ∪ B) = n(A) + n(B) – n(A ∩ B) = (3 + 6 – 3) = 6; n(A ∪ B)min = 6.
Universal Set
A non-empty set of which all the sets under consideration are subsets is called the universal set. It is usually denoted by ‘U’.
Complementary Set
The complement of a set A with respect to the Universal Set U is difference of U and A. Complement of set A is denoted by A (or Ac) (or A′).
A = U − A = {x : x ∈ U and x ∉ A}
we can say that A ∪ A = U (Universal Set) and A ∩ A = ϕ (Void Set)
Some of the useful properties/operations on sets are as follows
1. A ∪ U = U
2. A ∩ ϕ = ϕ
3. ϕc = U
4. Uc = ϕ
5. (A ∪ B)′ = A′ ∩ B′
6. (A ∩ B)′ = A′ ∪ B′
Power Set
The set of all subsets of a set is called the power set, denoted by P(A).
P(A) = {S : S ⊆ A}
It follows that P (A) contains 2n elements.
VENN DIAGRAMS
The diagram drawn to represent sets are called Venn diagrams or Eule–Venn diagrams. Here we represent the universal set U by points within rectangle and the subset A of the set U represented by the interior of a circle. If a set A is a subset of a set B then the circle representing A is drawn inside the circle representing B. If A and B are no equal but they have some common elements, then to represent A and B by two intersecting circles.
CARTESIAN PRODUCT OF SETS
Order pair: If a, b ∈ R, then (a, b) is known as order pair. i.e., pair of elements in which their order of existence is important. (a, b) ≠ (b, a).
Cartesian product: If A and B are any two sets, then the set of all order pairs from set A to set B is known as Cartesian Product and denoted by A × B and is defined as A × B = {(x, y) / x ∈ A and y ∈ B} e.g., A = {1, 2, 3}, B = {a, b}
A × B = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}
A × B ≠ B × A.
CARTESIAN PRODUCT OF SETS
Order pair: If a, b ∈ R, then (a, b) is known as order pair. i.e., pair of elements in which their order of existence is important. (a, b) ≠ (b, a).
Cartesian product: If A and B are any two sets, then the set of all order pairs from set A to set B is known as Cartesian Product and denoted by A × B and is defined as A × B = {(x, y) / x ∈ A and y ∈ B} e.g., A = {1, 2, 3}, B = {a, b}
A × B = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}
A × B ≠ B × A.
RELATIONSDefinition
Let A and B be two non-empty sets then every subset of A × B defines a relation from A to B and every relation from A to B is subset of A × B.
Let R ⊆ A × B and (a, b) ∈ R. Then we say that a is related to b by the relation R and write it as a R b. If (a, b) ∉ R, we write it as a ∃(a, b) ∉ R, a ∤R b.
Example
Let A = {1, 2, 3, 4, 5}, B = {1, 3}
We set a relation from A to B as: a R b iff a ≤ b; a ∈ A, b ∈ B. Then
R = {(1, 1), (1, 3), (2, 3), (3, 3)} ⊆ A × B
Domain and Range of a Relation
Let R be a relation from A to B, that is, let R ⊆ A × B. Then
Domain R = {a: a ∈ A, (a, b) ∈ R for some b ∈ B}
Also Range R = {b: b ∈ B, (a, b) ∈ R for some a ∈ A},
Thus Dom. R ⊆ A, Range R ⊆ B.
Total Number of Distinct Relations from A to B
Suppose the set A has m elements and the set B has n elements. Then A × B has 2mn different subsets which are different relations from A to B.
Inverse Relation
Let R ⊆ A × B be a relation from A to B. Then inverse relation R−1 ⊆ B × A is defined by
R−1 = {(b, a): (a, b) ∈ R, a ∈ A, b ∈ B}. It is clear that
- a R b ⇔ b R−1 a
- dom R−1 = range R and range R−1 = dom R
- (R−1)−1 = R
Example: Let A = {1, 2, 3, 4}, B = {a, b, c} and R = {(1, a), (1, c), (2, a)}. Then
(i) dom R = {1, 2}, range R = {a, c}
(ii) R−1 = {(a, 1), (c, 1), (a, 2)}
Compositions of Relations
Let R ⊆ A × B, S ⊆ B × C be two relations. Then compositions of the relations R and S denoted by S∘R ⊆ A × C and is defined by (a, c) ∈ (S∘R) iff ∃ b ∈ B such that (a, b) ∈ R, (b, c) ∈ S.
Reflexive Relation
R is a reflexive relation if (a, a) ∈ R, ∀ a ∈ A. It should be noted if there is at least one element a ∈ A such that (a, a) ∉ R, then R is not reflexive.
Example:
Let A = {1, 2, 3, 4, 5}
R = {(1, 1), (3, 2), (4, 2), (4, 4), (5, 2), (5, 5)} is not reflexive because 3 ∈ A and (3, 3) ∉ R.
Symmetric Relation
R is called a symmetric relation on A if (x, y) ∈ R ⇒ (y, x) ∈ R. That is, y R x whenever x R y.
It should be noted that R is symmetric iff R−1 = R. Let A = {1, 2, 3}, then R = {(1, 1), (1, 3), (3, 1)} is symmetric.
Anti-symmetric Relation
R is called an anti-symmetric relation if (a, b) ∈ R and (b, a) ∈ R ⇒ a = b. Thus, if a ≠ b then a may be related to b or b may be related to a, but never both. Or, we have never both a R b and b R a except when a = b.
Example: Let ℕ be the set of natural numbers. A relation R ⊆ ℕ × ℕ is defined by x R y iff x divides y (i.e. x/y).
Then x R y, y R x ⇒ x divides y, y divides x ⇒ x = y.
Transitive Relation
R is called a transitive relation if (a, b) ∈ R, (b, c) ∈ R ⇒ (a, c) ∈ R.
In other words, if a is related to b, b is related to c, then a is related to c.
Transitivity fails only when there exists a, b, c such that a R b, b R c but a not R c.
Equivalence Relation
A relation R in a set A is called an equivalence relation if
(i) R is reflexive i.e., (a, a) ∈ R, ∀ a ∈ A
(ii) R is symmetric i.e., (a, b) ∈ R ⇒ (b, a) ∈ R
(iii) R is transitive i.e., (a, b), (b, c) ∈ R ⇒ (a, c) ∈ R
The equivalence relation is usually denoted by the symbol ~.
Formulae and Concepts
1. A set is a collection of well defined objects which are distinct from each other.
2. Let A be any set. The set of all subsets of A is called power set of A and is denoted by P(A).
3. Power set of a given set is always non empty.
4. If A has n elements, then P(A) has 2ⁿ elements.
5. Equal sets are always equivalent but equivalent sets may not be equal.
6. n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C).
7. n(A − B) = n(A) − n(A ∩ B).
8. (A ∪ B)’ = A’ ∩ B’.
9. (A ∩ B)’ = A’ ∪ B’.
10. R is a reflexive relation if (a, a) ∈ R, ∀ a ∈ A.
11. R is called symmetric relation on A if (x, y) ∈ R ⇒ (y, x) ∈ R.
12. R is called anti-symmetric relation if (a, b) ∈ R and (b, a) ∈ R ⇒ a = b.
13. R is called transitive relation if (a, b) ∈ R, (b, c) ∈ R ⇒ (a, c) ∈ R.
Frequently Asked Questions
In Class 11 Mathematics, a set is defined as a well-defined collection of distinct objects. These objects can be numbers, alphabets, or even real-life entities like fruits, students, or books. For example, the set of first five natural numbers can be written as A = {1, 2, 3, 4, 5}.
The concept of sets forms the foundation of higher mathematics including algebra, probability, and relations. In Class 11, students learn how to represent sets in roster form (listing elements directly) and set-builder form (describing the rule of membership).
A set must follow two rules:
- Well-defined – Every element must be clearly identifiable. For example, “the set of vowels in English” = {a, e, i, o, u}.
- No repetition – Each element appears only once.
Practical Insight: In real life, sets are useful in organizing data. For instance, the set of “students who play football” can be compared with “students who play cricket” to find common players (intersection).
Table: Common Types of Sets
|
Type |
Example |
Explanation |
|
Finite Set |
{1, 2, 3} |
Countable elements |
|
Infinite Set |
{1, 2, 3…} |
Goes on endlessly |
|
Null/Empty Set |
{} |
No elements |
|
Universal Set |
U |
Contains all elements under consideration |
There are multiple types of sets in mathematics, each with unique properties. The most important include:
- Finite Set – A set with a limited number of elements. Example: A = {2, 4, 6}.
- Infinite Set – A set with unlimited elements. Example: N = {1, 2, 3, …}.
- Null/Empty Set (Ø) – Contains no elements. Example: C = {}.
- Singleton Set – Contains exactly one element. Example: D = {5}.
- Universal Set (U) – Contains all elements relevant to the discussion. Example: U = {1, 2, 3, 4, 5}.
- Subset – If every element of set A is in set B, A ⊆ B.
- Proper Subset – A ⊂ B but A ≠ B.
- Disjoint Sets – Sets with no common element.
Why It Matters: Understanding types of sets helps students master operations like union, intersection, and difference. In Class 11 exams, questions often ask to identify or classify sets, making this a high-yield topic.
Tip for Students: Always draw a Venn diagram when dealing with subsets or disjoint sets — it makes the concept much easier to visualize.
- Null/Empty Set (Ø): A set with no elements. Example: The set of natural numbers less than 1.
- Void Set: Another name for an empty set, symbolized by Ø or {}. Both terms mean the same thing.
- Universal Set (U): The “master set” that contains all possible elements for a particular discussion. For example, if we are studying natural numbers, then U = {1, 2, 3, …}.
Key Difference:
- Null set = “nothing”
- Universal set = “everything under consideration”
Table: Null vs Universal Set
|
Feature |
Null Set (Ø) |
Universal Set (U) |
|
Elements |
None |
All relevant elements |
|
Example |
{ } |
{1, 2, 3, 4, …, 100} |
|
Symbol |
Ø or {} |
U |
|
Role |
Represents impossibility |
Represents the scope of discussion |
Practical Example:
If U = {1, 2, 3, 4, 5}, and A = {2, 4}, then:
- A’s complement = {1, 3, 5}
- Ø represents a set like “numbers less than 0 in U”
This distinction is tested often in exams — especially in Venn diagram questions.
The laws of sets are mathematical rules that describe how sets interact during operations like union, intersection, and complement. They help simplify complex set expressions and solve exam problems efficiently.
The main laws include:
- Idempotent Law: A ∪ A = A, and A ∩ A = A
- Commutative Law: A ∪ B = B ∪ A, and A ∩ B = B ∩ A
- Associative Law: (A ∪ B) ∪ C = A ∪ (B ∪ C)
- Distributive Law: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
- Identity Law: A ∪ Ø = A, and A ∩ U = A
- Complement Law: A ∪ A’ = U, and A ∩ A’ = Ø
- De Morgan’s Laws: (A ∪ B)’ = A’ ∩ B’, and (A ∩ B)’ = A’ ∪ B’
Why This Matters:
These laws are the foundation for solving set-related proofs, simplifying expressions, and drawing Venn diagrams. They are also important in computer science (logic, databases, and programming).
A Venn diagram is a visual way of representing sets and their relationships using circles. Each circle represents a set, and overlaps show intersections (common elements).
- Union (A ∪ B): Shaded region covering both circles.
- Intersection (A ∩ B): Only the overlapping region.
- Difference (A – B): Elements of A that are not in B.
- Complement (A’): Everything in the universal set except A.
Example:
If U = {1,2,3,4,5,6}, A = {2,4,6}, and B = {1,2,3}, then:
- A ∪ B = {1,2,3,4,6}
- A ∩ B = {2}
- A – B = {4,6}
Why Helpful: Students easily grasp abstract set operations when they see them visually. For exams, most questions about unions and intersections can be solved faster with Venn diagrams.
De Morgan’s Laws explain how complements work with unions and intersections:
- (A ∪ B)’ = A’ ∩ B’
- (A ∩ B)’ = A’ ∪ B’
Explanation with Example:
If U = {1,2,3,4,5}, A = {1,2}, and B = {2,3}, then:
- A ∪ B = {1,2,3}, so (A ∪ B)’ = {4,5}.
- A’ = {3,4,5}, B’ = {1,4,5}, so A’ ∩ B’ = {4,5} = (A ∪ B)’. Verified!
Why It Matters:
- Crucial in Boolean algebra, probability, and computer logic.
- Commonly tested in Class 11 exams and competitive exams like JEE.
Yes, Set Theory is very important for JEE preparation. While the weightage is not as high as calculus or algebra, sets act as the foundation for Probability, Relations, and Functions topics that carry significant weight.
Why Important for JEE:
- Basics for Advanced Topics: Questions in probability, relations, and functions rely on set operations.
- Quick Scoring: Set theory problems are usually straightforward, making it a high-scoring chapter.
- Application in Logic: Helps in solving reasoning-type problems and advanced math.
Exam Strategy:
- Master definitions and operations.
- Practice previous year JEE questions on set operations, Venn diagrams, and De Morgan’s laws.
- Focus on universal sets and subsets, as these concepts often reappear in relation and function problems.
In the CBSE Class 11 Mathematics NCERT book, Chapter 1 is Sets. This chapter introduces the concept of sets, their representation, types, and operations.
Key Topics Covered in Chapter 1:
- Definition and representation of sets
- Types of sets (finite, infinite, null, singleton, universal)
- Subsets and supersets
- Power set and cardinality
- Venn diagrams
- Union, intersection, complement, and difference of sets
- Laws of sets (including De Morgan’s laws)
Why Chapter 1 Matters:
- It builds the base for later chapters like Relations & Functions, Probability, and Algebra.
- Questions are often direct and simple, making it a scoring area.
Study Strategy:
- Read examples and solved exercises carefully.
- Practice drawing Venn diagrams for clarity.
- Revise short notes before exams.
Class 11 introduces students to different classifications of sets, which include:
- Finite Set
- Infinite Set
- Null/Empty Set
- Singleton Set
- Equal Sets
- Equivalent Sets
- Subsets
- Proper Subsets
- Universal Set
- Power Set
- Disjoint Sets
- Overlapping Sets
Why It Matters:
This classification is crucial for exams — many MCQs and short-answer questions ask to identify or define set types.
Tip: Create a summary chart of set types with examples and practice at least 5–10 classification problems from NCERT and past papers.
Class 11 Mathematics is considered more challenging than Class 10 because it introduces abstract concepts, advanced algebra, and calculus foundations. The jump in difficulty often surprises students.
Why It Feels Difficult:
- More theoretical and logical reasoning required.
- Multiple new topics like Sets, Relations, Probability, Complex Numbers.
- Requires higher problem-solving stamina.
How to Manage:
- Strong Basics: Revise Class 10 algebra and geometry.
- Regular Practice: Solve NCERT examples before attempting reference books.
- Time Management: Divide topics into “easy scoring” (sets, statistics) and “conceptual heavy” (trigonometry, calculus).
- Consistency: 1–2 hours of daily math keeps learning smooth.
Motivation Insight: The difficulty curve flattens after consistent practice. Students who keep up with Class 11 Maths find Class 12 much easier to handle.
The hardest chapters vary for students, but most agree that Permutations & Combinations, Probability, and Trigonometric Functions are challenging.
- Permutations & Combinations: Needs logical thinking, not just formulas.
- Probability: Requires set theory understanding and application.
- Trigonometry: Involves identities and proofs.
Strategy to Overcome:
- Practice questions from NCERT and past board papers.
- Break large problems into smaller steps.
- Use visualization (trees, diagrams) for probability.
Scoring full marks in Class 11 Maths is achievable with the right strategy:
- Master NCERT First: Most board questions are NCERT-based.
- Revise Formulas Daily: Keep a separate formula notebook.
- Practice Step-by-Step Solutions: Show working in exams for partial credit.
- Solve PYQs: Understand question trends.
- Time Trials: Attempt sample papers within the exam duration.
Mindset Tip: Treat Maths like a sport — the more you practice, the sharper you become.