What is a set in mathematics?

A set is a well-defined collection of distinct objects called its elements. Well-defined means that for any object one can decide whether it belongs to the set or not. Sets are usually denoted by capital letters and elements by lowercase letters. Example: A = {1, 2, 3}.

What is the difference between roster form and set-builder form?

Roster form lists all elements inside braces, like A = {2, 4, 6}. Set-builder form gives a rule that describes the elements, like A = {x : x is an even natural number, x < 8}. Roster form is best for small finite sets, set-builder form for infinite or rule-based sets.

What is the formula for n(A union B)?

For any two finite sets A and B, n(A ∪ B) = n(A) + n(B) − n(A ∩ B). If A and B are disjoint (no common element), n(A ∩ B) = 0, so n(A ∪ B) = n(A) + n(B). This is a standard JEE inclusion-exclusion identity.

How many subsets does a set with n elements have?

A set with n elements has exactly 2^n subsets, including the empty set and the set itself. Of these, 2^n − 1 are proper subsets. The collection of all subsets is called the power set, denoted P(A).

What are De Morgan's laws for sets?

De Morgan's laws state that the complement of a union equals the intersection of complements, and the complement of an intersection equals the union of complements: (A ∪ B)^c = A^c ∩ B^c and (A ∩ B)^c = A^c ∪ B^c. They are frequently used to simplify set expressions in JEE problems.

What is the difference between phi and {phi}?

The empty set φ has zero elements: n(φ) = 0. The set {φ} contains one element (the empty set itself), so n({φ}) = 1. This is a common trick question in JEE, so watch the braces carefully.

Is A × B always equal to B × A?

No. A × B and B × A both have the same number of elements (mn if |A| = m and |B| = n), but their elements are ordered pairs in a specific order. So (a, b) ∈ A × B and (b, a) ∈ B × A are different. A × B = B × A only when A = B.

Why is the empty set a subset of every set?

By definition, A ⊆ B means every element of A is also in B. Since φ has no elements, this condition is vacuously true for any set B - there is nothing in φ that could fail to be in B. Hence φ ⊆ B for every set B.

Set

Q: What is the formula for n(A union B)?

For any two finite sets A and B, n(A ∪ B) = n(A) + n(B) − n(A ∩ B). If A and B are disjoint (no common element), n(A ∩ B) = 0, so n(A ∪ B) = n(A) + n(B). This is a standard JEE inclusion-exclusion identity.

Q: How many subsets does a set with n elements have?

A set with n elements has exactly 2^n subsets, including the empty set and the set itself. Of these, 2^n − 1 are proper subsets. The collection of all subsets is called the power set, denoted P(A).

Q: What are De Morgan's laws for sets?

De Morgan's laws state that the complement of a union equals the intersection of complements, and the complement of an intersection equals the union of complements: (A ∪ B)^c = A^c ∩ B^c and (A ∩ B)^c = A^c ∪ B^c. They are frequently used to simplify set expressions in JEE problems.

Q: What is the difference between phi and {phi}?

The empty set φ has zero elements: n(φ) = 0. The set {φ} contains one element (the empty set itself), so n({φ}) = 1. This is a common trick question in JEE, so watch the braces carefully.

Q: Is A × B always equal to B × A?

No. A × B and B × A both have the same number of elements (mn if |A| = m and |B| = n), but their elements are ordered pairs in a specific order. So (a, b) ∈ A × B and (b, a) ∈ B × A are different. A × B = B × A only when A = B.

Q: Why is the empty set a subset of every set?

By definition, A ⊆ B means every element of A is also in B. Since φ has no elements, this condition is vacuously true for any set B - there is nothing in φ that could fail to be in B. Hence φ ⊆ B for every set B.

MathsSets, Relations And Number SystemFor JEE aspirants

A set is a well-defined collection of distinct objects. In JEE mathematics, sets form the foundation for functions, relations, probability, and coordinate geometry. This page covers everything you need on sets - definition, representations (roster and set-builder), types of sets (empty, singleton, finite, infinite, equal, disjoint), operations (union, intersection, difference, symmetric difference), subset and power set, universal set and complement, the number-of-elements formulas for two and three sets, algebra of sets, and Cartesian product - with worked examples in JEE style.

Key Formulas - Quick Reference

Number of subsets of a set with $n$ elements: $2^{n}$
Number of proper subsets: $2^{n} - 1$
Power set: $P (A) = {S : S \subseteq A}$ , with $∣ P (A) ∣ = 2^{n}$
$n (A \cup B) = n (A) + n (B) - n (A \cap B)$
$n (A \cup B \cup C) = n (A) + n (B) + n (C) - n (A \cap B) - n (B \cap C) - n (A \cap C) + n (A \cap B \cap C)$
$n (A - B) = n (A) - n (A \cap B)$
De Morgan's laws: $(A \cup B)^{c} = A^{c} \cap B^{c}$ and $(A \cap B)^{c} = A^{c} \cup B^{c}$
If $∣ A ∣ = m$ and $∣ B ∣ = n$ , then $∣ A \times B ∣ = mn$

1. What is a Set?

A set is a well-defined collection of distinct objects. "Well-defined" means there is a rule that decides whether any given object belongs to the collection or not. Sets are usually denoted by capital letters $A, B, C, X, Y, Z$ , and their elements by lowercase letters. If $x$ is an element of set $A$ , we write $x \in A$ ; otherwise $x \in / A$ .

Representation of a Set

There are two standard ways to represent a set.

Roster form (tabular form): List all elements inside braces separated by commas.
Example: The set of even natural numbers less than 10 is

A = {2, 4, 6, 8}

Set-builder form (rule form): Write a variable followed by the property each element satisfies.
Example:

A = {x : x is a natural number and x < 10}

A = {x ∣ x \in N, x < 10}

2. Types of Sets

Null (Empty / Void) Set

A set with no element is called the null set or empty set, denoted by $ϕ$ or ${}$ . The number of elements of set $A$ is $n (A)$ , and $n (ϕ) = 0$ .

Singleton Set

A set containing exactly one element, e.g. ${5}$ or ${a}$ .

Finite and Infinite Sets

A set with a finite number of elements is finite; otherwise it is infinite. Example: the set of days in a week is finite; the set of natural numbers $N$ is infinite.

Equal Sets

Sets $A$ and $B$ are equal if $A \subseteq B$ and $B \subseteq A$ . We write $A = B$ , meaning both have exactly the same elements.

Disjoint Sets

If $A \cap B = ϕ$ , the sets $A$ and $B$ share no common element and are called disjoint sets.

3. Set Operations

Union of Sets

The union of two or more sets is the set of all elements that belong to any of these sets. Symbol: $\cup$ .

$A \cup B = {x : x \in A or x \in B}$

Example: If $A = {1, 2, 3, 4}$ , $B = {2, 4, 5, 6}$ , $C = {1, 2, 6, 8}$ , then $A \cup B \cup C = {1, 2, 3, 4, 5, 6, 8}$ .

Figure: Union of sets A and B - all elements that belong to A or B.

Intersection of Sets

The intersection is the set of elements common to all the sets. Symbol: $\cap$ .

$A \cap B = {x : x \in A and x \in B}$

Example: If $A = {1, 2, 3, 4}$ , $B = {2, 4, 5, 6}$ , $C = {1, 2, 6, 8}$ , then $A \cap B \cap C = {2}$ .

Figure: Intersection of sets A and B - elements common to both.

Difference of Sets

$A - B$ (or $A ∖ B$ ) is the set of elements that are in $A$ but not in $B$ .

$A - B = {x : x \in A and x \in / B}$

Similarly $B - A = {x : x \in B and x \in / A}$ . In general, $A - B \neq = B - A$ .

Example: If $A = {a, b, c, d}$ and $B = {b, c, e, f}$ , then $A - B = {a, d}$ and $B - A = {e, f}$ .

Figure: Set difference A − B - part of A that lies outside B.

Symmetric Difference

The symmetric difference of $A$ and $B$ is $A Δ B = (A - B) \cup (B - A) .$ It contains the elements in exactly one of the two sets.

Figure: Symmetric difference A Δ B - elements in exactly one of A or B.

4. Subset and Power Set

Subset

$A$ is a subset of $B$ if every element of $A$ is also in $B$ . Symbol: $A \subseteq B$ .

Every set is a subset of itself. The empty set $ϕ$ is a subset of every set. If $A \subseteq B$ and there is at least one element in $B$ that is not in $A$ , then $A$ is a proper subset: $A \subset B$ .

Figure: A is a subset of B - every element of A lies inside B.

Power Set

The power set of $A$ , denoted $P (A)$ , is the set of all subsets of $A$ . If $∣ A ∣ = n$ , then $∣ P (A) ∣ = 2^{n}$ .

Example: If $A = {a, b}$ , then $P (A) = {ϕ, {a}, {b}, {a, b}}$ , so $∣ P (A) ∣ = 4 = 2^{2}$ .

5. Universal Set and Complement

Universal Set

A non-empty set of which all sets under consideration are subsets is called the universal set, denoted by $U$ . For example, when working with operations on real numbers, $U = R$ .

Complementary Set

The complement of $A$ with respect to $U$ is $A^{c}$ (or $\overset{ˉ}{A}$ or $A^{'}$ ).

$A^{c} = U - A = {x : x \in U and x \in / A}$

Important identities: $A \cup A^{c} = U$ , $A \cap A^{c} = ϕ$ , $ϕ^{c} = U$ , $U^{c} = ϕ$ .

6. Formulas for Number of Elements

Let $A$ , $B$ , $C$ be finite sets and $U$ a finite universal set. Then:

$n (A \cup B) = n (A) + n (B) - n (A \cap B)$
If $A$ and $B$ are disjoint, $n (A \cup B) = n (A) + n (B)$
$n (A - B) = n (A) - n (A \cap B)$ , so $n (A) = n (A - B) + n (A \cap B)$
$n (A \cup B \cup C) = n (A) + n (B) + n (C) - n (A \cap B) - n (B \cap C) - n (A \cap C) + n (A \cap B \cap C)$
Elements in exactly two of $A, B, C$ : $n (A \cap B) + n (B \cap C) + n (C \cap A) - 3 n (A \cap B \cap C)$
Elements in at least two of $A, B, C$ : $n (A \cap B) + n (A \cap C) + n (B \cap C) - 2 n (A \cap B \cap C)$
Elements in exactly one of $A, B, C$ : $n (A) + n (B) + n (C) - 2 n (A \cap B) - 2 n (B \cap C) - 2 n (A \cap C) + 3 n (A \cap B \cap C)$

7. Algebra of Sets

These identities are the workhorse tools for simplifying set expressions.

Law	Statement
Idempotent	$A \cup A = A$ ; $A \cap A = A$
Identity	$A \cup ϕ = A$ ; $A \cap U = A$
Commutative	$A \cup B = B \cup A$ ; $A \cap B = B \cap A$
Associative	$(A \cup B) \cup C = A \cup (B \cup C)$ ; $(A \cap B) \cap C = A \cap (B \cap C)$
Distributive	$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ ; $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
De Morgan's	$(A \cup B)^{c} = A^{c} \cap B^{c}$ ; $(A \cap B)^{c} = A^{c} \cup B^{c}$
Involution	$(A^{c})^{c} = A$
Domination	$A \cup U = U$ ; $A \cap ϕ = ϕ$

8. Cartesian Product of Sets

The Cartesian product (or cross product) of sets $A$ and $B$ , written $A \times B$ , is the set of all ordered pairs $(x, y)$ with $x \in A$ and $y \in B$ . Since the pair $(a, b)$ is not the same as $(b, a)$ unless $a = b$ , we have $A \times B \neq = B \times A$ in general.

$A \times B = {(x, y) : x \in A and y \in B}$

If $∣ A ∣ = m$ and $∣ B ∣ = n$ , then $∣ A \times B ∣ = mn$ . The definition extends to more than two sets: $A_{1} \times A_{2} \times \dots \times A_{n} = {(a_{1}, a_{2}, \dots, a_{n}) : a_{i} \in A_{i} for each i} .$

Solved Examples

Solved Example 1

A

and

B

are two sets containing 3 and 6 elements respectively, find the minimum and maximum number of elements in

A \cup B

Solution:

Using $n (A \cup B) = n (A) + n (B) - n (A \cap B)$ , the size of the union is maximum when $n (A \cap B)$ is minimum, and minimum when $n (A \cap B)$ is maximum.

Case 1 (minimum intersection): $A \cap B = ϕ$ , so $n (A \cap B) = 0$ . Then $n (A \cup B) = 3 + 6 - 0 = 9$ . Hence $n (A \cup B)_{m a x} = 9$ .

Case 2 (maximum intersection): This happens when $A \subseteq B$ , giving $n (A \cap B) = 3$ . Then $n (A \cup B) = 3 + 6 - 3 = 6$ . Hence $n (A \cup B)_{m i n} = 6$ .

So $6 \leq n (A \cup B) \leq 9$ .

Solved Example 2

In a group of 1000 people, 750 can speak Hindi and 400 can speak Bengali. How many can speak Hindi only? How many can speak both?

Solution:

Let $H$ = Hindi speakers, $B$ = Bengali speakers. Given: $n (H \cup B) = 1000$ , $n (H) = 750$ , $n (B) = 400$ .

$n (H \cap B) = n (H) + n (B) - n (H \cup B) = 750 + 400 - 1000 = 150$ . So 150 can speak both.

Hindi only: $n (H) - n (H \cap B) = 750 - 150 = 600$ . So 600 speak Hindi only.

Solved Example 3

A survey shows 63% of Americans like cheese and 76% like apples. If

x %

like both, find the range of

x

Solution:

Let $A$ = cheese lovers, $B$ = apple lovers, expressed as percentages of the total. $n (A) = 63$ , $n (B) = 76$ .

Since $n (A \cup B) \leq 100$ , using $n (A \cap B) = n (A) + n (B) - n (A \cup B)$ :

$x = 63 + 76 - n (A \cup B) \geq 63 + 76 - 100 = 39$ .

Also $n (A \cap B) \leq min (n (A), n (B)) = 63$ , so $x \leq 63$ .

Therefore $39 \leq x \leq 63$ .

Solved Example 4

A class has 175 students. The following table shows the number of students studying one or more of these subjects:
Mathematics = 100, Physics = 70, Chemistry = 46,
Mathematics and Physics = 30, Mathematics and Chemistry = 28, Physics and Chemistry = 23,
Mathematics, Physics and Chemistry = 18.
How many students study Mathematics alone, Physics alone, and Chemistry alone? Are there students who have not offered any of these subjects?

Solution:

Let $M$ , $P$ , $C$ denote the sets of students studying Mathematics, Physics, Chemistry respectively. Given:

$n (M) = 100$ , $n (P) = 70$ , $n (C) = 46$ , $n (M \cap P) = 30$ , $n (M \cap C) = 28$ , $n (P \cap C) = 23$ , $n (M \cap P \cap C) = 18$ .

Total studying at least one subject: By inclusion-exclusion,

$n (M \cup P \cup C) = n (M) + n (P) + n (C) - n (M \cap P) - n (M \cap C) - n (P \cap C) + n (M \cap P \cap C) .$

$n (M \cup P \cup C) = 100 + 70 + 46 - 30 - 28 - 23 + 18 = 153.$

Students not offered any subject: $175 - 153 = 22$ . Yes, there are 22 such students.

Mathematics alone: $n (M) - n (M \cap P) - n (M \cap C) + n (M \cap P \cap C) = 100 - 30 - 28 + 18 = 60$ .

Physics alone: $n (P) - n (M \cap P) - n (P \cap C) + n (M \cap P \cap C) = 70 - 30 - 23 + 18 = 35$ .

Chemistry alone: $n (C) - n (M \cap C) - n (P \cap C) + n (M \cap P \cap C) = 46 - 28 - 23 + 18 = 13$ .

Solved Example 5

A = {a, b, c}

and

B = {b, c, d}

, evaluate: (i)

A \cup B

A \cap B

A - B

B - A

; (ii)

A \times B

and

B \times A

Solution:

(i)

$A \cup B = {a, b, c, d}$

$A \cap B = {b, c}$

$A - B = {a}$

$B - A = {d}$

(ii)

$A \times B = {(a, b), (a, c), (a, d), (b, b), (b, c), (b, d), (c, b), (c, c), (c, d)}$

$B \times A = {(b, a), (b, b), (b, c), (c, a), (c, b), (c, c), (d, a), (d, b), (d, c)}$

Note that $A \times B \neq = B \times A$ , though both have $3 \times 3 = 9$ elements.

Common Mistakes to Avoid

Watch out

Confusing $\in$ (element of) with $\subseteq$ (subset of). Example: $1 \in {1, 2}$ but ${1} \subseteq {1, 2}$ .
Treating $ϕ$ and ${ϕ}$ as the same. The first has $0$ elements; the second has $1$ element (the empty set itself).
Forgetting that $A - B \neq = B - A$ in general.
Applying $n (A \cup B) = n (A) + n (B)$ without checking disjointness - this only holds when $A \cap B = ϕ$ .
Assuming $A \times B = B \times A$ . They have the same cardinality but are equal only if $A = B$ .
Missing $ϕ$ as a subset when listing all subsets of a small set.

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