Set Operations : Union, Intersection, Disjoint, Complement, Difference and Symmetric Difference of Sets

Set Operations are a core part of Class 11 CBSE Mathematics and are essential for understanding relationships between sets. In this topic, we study operations such as union, intersection, difference, complement, and symmetric difference, along with special cases like disjoint sets. These operations help in solving problems using logical reasoning and are widely applied in algebra, probability, and competitive exams.

What are Basic Operations On Sets ?

Set Operations can be defined as the operations performed on two or more sets to obtain a single set containing a combination of elements from all the sets being operated upon.

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Union of Sets in Math – Definition, Solved Examples

The union of two sets $A$ and $B$, denoted by $A \cup B$, is the set of all those elements which are either in $A$ or in $B$ or in both $A$ and $B$.

Thus,
$$
A \cup B = \{ x : x \in A \text{ or } x \in B \}
\Leftrightarrow x \in A \cup B \Leftrightarrow x \in A \text{ or } x \in B.
$$

Example: 

If A = {2, 4, 8} and B = {2, 6, 8}
then the union of A and B is the set A ∪ B = {2, 4, 6, 8}

In this example, 2, 4, 6, and 8 are the elements that are found in set A or in set B or in both sets A and B

Solved Problems based on Union of Sets

EXAMPLE 1
If $A = \{3, 4, 5, 6\}$ and $B = \{4, 6, 8, 10\}$, find $A \cup B$.

SOLUTION
Clearly,
$$
A \cup B = \{3, 4, 5, 6, 8, 10\}.
$$

EXAMPLE 2
Let $A = \{x : x \text{ is a prime number less than } 10\}$ and
$B = \{x : x \in N, x \text{ is a factor of } 12\}$. Find $A \cup B$.

SOLUTION
We have
$$
A = \{2, 3, 5, 7\} \quad \text{and} \quad B = \{1, 2, 3, 4, 6, 12\}.
$$
$\therefore A \cup B = \{1, 2, 3, 4, 5, 6, 7, 12\}$.

EXAMPLE 3
Let $A = \{x : x \text{ is a positive integer}\}$ and
let $B = \{x : x \text{ is a negative integer}\}$. Find $A \cup B$.

SOLUTION
Clearly,
$$
A \cup B = \{x : x \text{ is an integer and } x \neq 0\}.
$$

EXAMPLE 4
If $A = \{x : x = 2n + 1, n \in Z\}$ and
$B = \{x : x = 2n, n \in Z\}$ then find $A \cup B$.

SOLUTION
We have
$
A \cup B = \{x : x \text{ is an odd integer}\} \cup \{x : x \text{ is an even integer}\}
= \{x : x \text{ is an integer}\} = Z.
$

Key Point
The union of $n$ sets $A_1, A_2, A_3, \dots, A_n$ is denoted by $$(A_1 \cup A_2 \cup A_3 \cup \dots \cup A_n)=\bigcup_{i=1}^{n} A_i.$$

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Intersection Of Sets in Math – Definition, Solved Examples

The intersection of two sets $A$ and $B$, denoted by $A \cap B$, is the set of all elements which are common to both $A$ and $B$.
Thus,
$$A \cap B = \{ x : x \in A \text{ and } x \in B \}.$$

$\therefore\ x \in A \cap B \Leftrightarrow x \in A$ and $x \in B$
or
$x \in A \cap B \Leftrightarrow x \in A\ \&\ x \in B.$

Example: 

If A = {2, 3, 5, 7} and B = {1, 2, 3, 4, 5}
then the intersection of set A and B is the set A ∩ B = {2, 3, 5} 

In this example 2, 3, and 5 are the only elements that belong to both sets A and B. 

Solved Problems based on Intersection Of Sets

EXAMPLE 5
Let $A = \{1, 3, 5, 7, 9\}$ and $B = \{2, 3, 5, 7, 11, 13\}$. Find $A \cap B$.

SOLUTION
We have
$$
A \cap B = \{1, 3, 5, 7, 9\} \cap \{2, 3, 5, 7, 11, 13\} = \{3, 5, 7\}.
$$

EXAMPLE 6
If $A = \{x : x \in \mathbb{N}, x \text{ is a factor of } 12\}$
and $B = \{x : x \in \mathbb{N}, x \text{ is a factor of } 18\}$, find $A \cap B$.

SOLUTION
We have
$$
A = \{x : x \in \mathbb{N}, x \text{ is a factor of } 12\} = \{1, 2, 3, 4, 6, 12\},
$$

$$
B = \{x : x \in \mathbb{N}, x \text{ is a factor of } 18\} = \{1, 2, 3, 6, 9, 18\}.
$$

$$
\therefore \quad A \cap B = \{1, 2, 3, 6\}.
$$

EXAMPLE 7
If $A = \{x : x = 3n, n \in \mathbb{Z}\}$ and
$B = \{x : x = 4n, n \in \mathbb{Z}\}$ then find $A \cap B$.

SOLUTION
We have
$$
A = \{x : x \in \mathbb{Z} \text{ and } x \text{ is a multiple of } 3\}
$$

$$
B = \{x : x \in \mathbb{Z} \text{ and } x \text{ is a multiple of } 4\}.
$$

$$
\therefore \quad A \cap B = \{x : x \in \mathbb{Z} \text{ and } x \text{ is a multiple of both } 3 \text{ and } 4\}
$$

$$
A \cap B = \{x : x \in \mathbb{Z} \text{ and } x \text{ is a multiple of } 12\}
$$

$$
A \cap B = \{x : x = 12n, n \in \mathbb{Z}\}.
$$

Hence,

$$
A \cap B = \{x : x = 12n, n \in \mathbb{Z}\}.
$$

EXAMPLE 8
If $A = (2, 4)$ and $B = [3, 5)$, find $A \cap B$.

SOLUTION We have
$$
A = (2, 4) = \{x : x \in \mathbb{R}, 2 < x < 4\}
$$

$$
B = [3, 5) = \{x : x \in \mathbb{R}, 3 \leq x < 5\}
$$

Clearly,
$$
A \cap B = \{x : x \in \mathbb{R}, 3 \leq x < 4\} = [3, 4).
$$

Key Point
The intersection of $n$ sets $A_1, A_2, A_3, \dots, A_n$ is denoted by $$ (A_1 \cap A_2 \cap A_3 \cap \dots \cap A_n) = \bigcap_{i=1}^{n} A_i. $$

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Disjoint Sets – Definition, Solved Example

For any two sets A and B which do have no common elements are called Disjoint Sets. The intersection of the Disjoint set is ϕ. 

Two sets $A$ and $B$ are said to be disjoint if $A \cap B = \emptyset$.

Example:

Check whether Set A = {a, b, c, d} and Set B = {1, 2} are disjoint or not.
Set A ={a, b, c, d}
Set B= {1, 2}
Here, A ∩ B =  ϕ
Thus, Set A and Set B are disjoint sets.

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Intersecting Sets or Overlapping Sets – Definition, Solved Examples

Two sets $A$ and $B$ are said to be intersecting if $A \cap B \neq \emptyset$.

Solved Problems based on Intersecting Sets

EXAMPLE 9
If $A = \{1, 3, 5, 7, 9\}$, $B = \{2, 4, 6, 8\}$ and $C = \{2, 3, 5, 7, 11\}$,
find $(A \cap B)$ and $(A \cap C)$. What do you conclude?

SOLUTION
We have
$$
A \cap B = \{1, 3, 5, 7, 9\} \cap \{2, 4, 6, 8\} = \emptyset
$$

and

$$
A \cap C = \{1, 3, 5, 7, 9\} \cap \{2, 3, 5, 7, 11\} = \{3, 5, 7\} \neq \emptyset.
$$

Thus, $A$ and $B$ are disjoint sets while $A$ and $C$ are intersecting sets.

EXAMPLE 10
Give examples of three sets $A, B, C$ such that
$(A \cap B) \neq \emptyset$,
$(B \cap C) \neq \emptyset$,
$(A \cap C) \neq \emptyset$ and
$(A \cap B \cap C) = \emptyset$.

SOLUTION
Consider the sets $A = \{1, 2\}$, $B = \{2, 3, 4\}$ and $C = \{1, 3, 5\}$.

Then,
$$
(A \cap B) = \{1, 2\} \cap \{2, 3, 4\} = \{2\} \neq \emptyset;
$$

$$
(B \cap C) = \{2, 3, 4\} \cap \{1, 3, 5\} = \{3\} \neq \emptyset;
$$

$$
(A \cap C) = \{1, 2\} \cap \{1, 3, 5\} = \{1\} \neq \emptyset;
$$

$$
(A \cap B \cap C) = \{1, 2\} \cap \{2, 3, 4\} \cap \{1, 3, 5\} = \emptyset.
$$

Thus, $A \cap B \neq \emptyset$, $B \cap C \neq \emptyset$, $A \cap C \neq \emptyset$ and $A \cap B \cap C = \emptyset$.

EXAMPLE 11
Give an example of three sets $A, B, C$ such that
$A \cap B = A \cap C$ but $B \neq C$.

SOLUTION
Consider the sets $A = \{1, 2, 3\}$, $B = \{3, 4\}$ and $C = \{3, 5, 7\}$.

Then,
$$
A \cap B = \{1, 2, 3\} \cap \{3, 4\} = \{3\}.
$$

and,

$$
A \cap C = \{1, 2, 3\} \cap \{3, 5, 7\} = \{3\}.
$$

Thus, $A \cap B = A \cap C$ and clearly, $B \neq C$.

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Difference Of Sets – Definition, Solved Examples

If A and B are two sets, then their difference A – B is the set of all those elements of A which do not belong to B.

For any sets $A$ and $B$, their difference $(A – B)$ is defined as

$$
(A – B) = \{x : x \in A \text{ and } x \notin B\}.
$$

Thus,

$$
x \in (A – B) \Leftrightarrow x \in A \text{ and } x \notin B.
$$

Method : Identify the elements in set A$$ and remove any elements that also exist in set B.

$$Similarly, the difference B – A is the set of all those elements of B that do not belong to A. i.e.
B – A = { x : x ∈ B and x ∉ A}
x ∈ B – A ⇔ x ∈ B and x ∉ A

Note:  A – B is equivalent to A ∩ B’ i.e.,  A – B = A ∩ B’

Example: 

If A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and B = {2, 3, 5, 7}
then A – B = {1, 4, 6, 8, 9, 10}
and further B – A = ∅

Solved Problems based on Difference Of Sets

EXAMPLE 12
If $A = \{x : x \in \mathbb{N}, x \text{ is a factor of } 6\}$ and
$B = \{x : x \in \mathbb{N}, x \text{ is a factor of } 8\}$ then
find (i) $A \cup B$, (ii) $A \cap B$, (iii) $A – B$, (iv) $B – A$.

SOLUTION
We have
$$
A = \{x : x \in \mathbb{N}, x \text{ is a factor of } 6\} = \{1, 2, 3, 6\}
$$

and

$$
B = \{x : x \in \mathbb{N}, x \text{ is a factor of } 8\} = \{1, 2, 4, 8\}.
$$

(i)
$$
A \cup B = \{1, 2, 3, 6\} \cup \{1, 2, 4, 8\} = \{1, 2, 3, 4, 6, 8\}.
$$

(ii)
$$
A \cap B = \{1, 2, 3, 6\} \cap \{1, 2, 4, 8\} = \{1, 2\}.
$$

(iii)
$$
A – B = \{1, 2, 3, 6\} – \{1, 2, 4, 8\} = \{3, 6\}.
$$

(iv)
$$
B – A = \{1, 2, 4, 8\} – \{1, 2, 3, 6\} = \{4, 8\}.
$$

Key Point
A – B ≠ B – A

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Symmetric Difference Of Sets – Definition, Solved Examples

The symmetric difference of two sets $A$ and $B$, denoted by $A \Delta B$, is defined as

$$
A \Delta B = (A – B) \cup (B – A).
$$

Key Point
A Δ B = { x : x ∉ A ∩ B}

Solved Problem based on Symmetric Difference Of Sets

EXAMPLE 13
Let $A = \{a, b, c, d\}$ and
$B = \{b, d, f, g\}$. Find $A \Delta B$.

SOLUTION
We have
$$
(A – B) = \{a, b, c, d\} – \{b, d, f, g\} = \{a, c\},
$$

$$
(B – A) = \{b, d, f, g\} – \{a, b, c, d\} = \{f, g\}.
$$

$$
\therefore \quad A \Delta B = \{a, c\} \cup \{f, g\} = \{a, c, f, g\}.
$$

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Complement Of A Set – Definition, Solved Examples

Let U be the universal set and A is a subset of U. Then, the complement of A with respect to (w. r. t) U is the set of all elements of U which are not the elements of A. Complement of A with respect to U is denoted by A’ or A^c.

Let $U$ be the universal set and let $A \subseteq U$. Then, the complement of $A$, denoted by $A’$ or $(U – A)$, is defined as
$$
A’ = \{x : x \in U \text{ and } x \notin A\}.
$$
Clearly,
$$
x \in A’ \Leftrightarrow x \in U \text{ and } x \notin A.
$$

For example :

If U = { 1, 2, 3, 4, 5, 6} and A = { 2, 4, 6}
then, A’ = U– A = {1, 2, 3, 4, 5, 6} – { 2, 4, 6} ⇒ A’ = { 1, 3, 5}

Solved Problem based on Complement Of A Set

EXAMPLE 14
If $U = \{1, 2, 3, 4, 5, 6, 7, 8\}$ and
$A = \{2, 4, 6, 8\}$, find (i) $A’$ (ii) $(A’)’$.

SOLUTION
We have

(i)
$$
A’ = U – A = \{1, 2, 3, 4, 5, 6, 7, 8\} – \{2, 4, 6, 8\} = \{1, 3, 5, 7\}.
$$

(ii)
$$
(A’)’ = U – A’ = \{1, 2, 3, 4, 5, 6, 7, 8\} – \{1, 3, 5, 7\} = \{2, 4, 6, 8\} = A.
$$

Key Point
(A‘)’ = A

EXAMPLE 15
Let $N$ be the universal set.
(i) If $A = \{x : x \in N \text{ and } x \text{ is odd}\}$, find $A’$.
(ii) If $B = \{x : x \in N, x \text{ is divisible by } 3 \text{ and } 5\}$, find $B’$.

SOLUTION
We have
(i)
$$
A’ = \{x : x \in \mathbb{N} \text{ and } x \text{ is not odd}\} = \{x : x \in \mathbb{N} \text{ and } x \text{ is even}\}.
$$

(ii)
$$
B’ = \{x : x \in \mathbb{N}, x \text{ is not divisible by } 3 \text{ or } x \text{ is not divisible by } 5\}.
$$

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Theorems Based On Complementation

EXAMPLE 16
If $A \subseteq U$, prove that:
(i) $U’ = \emptyset$
(ii) $\emptyset’ = U$
(iii) $(A’)’ = A$
(iv) $A \cup A’ = U$,
(v) $A \cap A’ = \emptyset$

SOLUTION
We have
(i)
$$
U’ = \{x : x \in \phi \text{ and } x \notin U\} = \emptyset.
$$

(ii)
$$
\phi’ = \{x : x \in U \text{ and } x \notin \phi\} = U.
$$

(iii)
$$
(A’)’ = \{x : x \in U \text{ and } x \notin A’\} = \{x : x \in U \text{ and } x \in A\} = A.
$$

(iv)
$$A \cup A’ = \{x : x \in U \text{ and } (x \in A \text{ or } x \in A’)\} $$

$$ A \cup A’= \{x : x \in U \text{ and } (x \in A \text{ or } x \notin A)\} = U.$$

A ∪ A ′ = {x ∈ U : x ∈ A} ∪ {x ∈ U : x ∉ A} = U

(v)
$$
A \cap A’ = \{x : x \in U \text{ and } x \in A \text{ and } x \in A’\}
$$

$$
A \cap A’ = \{x : x \in U \text{ and } x \in A \text{ and } x \notin A\} = \emptyset.
$$

A ∩ A ′ = {x ∈ U : x ∈ A} ∩ {x ∈ U : x ∉ A} = Φ

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FAQs Based On Operations on Sets

What is the union of two sets?

The union of two sets includes all elements that are in either set or both:
$$
A \cup B = {x : x \in A \text{ or } x \in B}
$$

What is the intersection of two sets?

The intersection contains only the common elements of both sets:
$$
A \cap B = {x : x \in A \text{ and } x \in B}
$$

What are disjoint sets?

Two sets are called disjoint if they have no common elements:
$$
A \cap B = \emptyset
$$

What is the complement of a set?

The complement of a set $A$ consists of all elements in the universal set $U$ that are not in $A$:
$$
A’ = U – A
$$

What is the difference between two sets?

The difference $A – B$ contains elements that are in $A$ but not in $B$:
$$
A – B = {x : x \in A \text{ and } x \notin B}
$$

What is symmetric difference of sets?

The symmetric difference includes elements that are in either set but not in both:
$$
A \triangle B = (A – B) \cup (B – A)
$$

What is the relation between union and intersection?

Union combines all elements, while intersection gives only common elements:
$$
A \cap B \subseteq A \cup B
$$

Can two sets be both disjoint and equal?

No, equal sets must have exactly the same elements, while disjoint sets have no common elements.

What happens if a set is united with itself?

The union of a set with itself gives the same set:
$$
A \cup A = A
$$

What happens if a set is intersected with itself?

The intersection of a set with itself also gives the same set:
$$
A \cap A = A
$$

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Practice Exercise based on Set Operations

1.
If $A = \{a, b, c, d, e, f\}$, $B = \{c, e, g, h\}$ and $C = \{a, e, m, n\}$, find:

(i) $A \cup B$ (ii) $B \cup C$ (iii) $A \cup C$

(iv) $B \cap C$ (v) $C \cap A$ (vi) $A \cap B$

2. If
$A = \{1, 2, 3, 4, 5\}$,
$B = \{4, 5, 6, 7, 8\}$,
$C = \{7, 8, 9, 10, 11\}$
and
$D = \{10, 11, 12, 13, 14\}$,
find:

(i) $A \cup B$ (ii) $B \cup C$ (iii) $A \cup C$

(iv) $B \cup D$ (v) $(A \cup B) \cup C$ (vi) $(A \cup B) \cap C$

(vii) $(A \cup B) \cap D$ (viii) $(A \cup B) \cap (B \cup C)$

(ix) $(A \cup C) \cap (C \cup D)$

3. If
$A = \{3, 5, 7, 9, 11\}$,
$B = \{7, 9, 11, 13\}$,
$C = \{11, 13, 15\}$
and
$D = \{15, 17\}$,
find:

(i) $A \cap B$ (ii) $A \cap C$ (iii) $B \cap C$

(iv) $B \cap D$ (v) $(B \cap C) \cap D$ (vi) $(A \cap B) \cap C$

4. If
$A = \{x : x \in \mathbb{N}\}$,
$B = \{x : x \in \mathbb{N} \text{ and } x \text{ is even}\}$,
$C = \{x : x \in \mathbb{N} \text{ and } x \text{ is odd}\}$
and
$D = \{x : x \in \mathbb{N} \text{ and } x \text{ is prime}\}$
then find:

(i) $A \cap B$ (ii) $A \cap C$ (iii) $A \cap D$

(iv) $B \cap C$ (v) $B \cap D$ (vi) $C \cap D$

5. If
$A = \{x : x \in \mathbb{N}\}$,
$B = \{(x+2) : x \in \mathbb{N} \text{ and } 1 \leq x < 4\}$,
$C = \{x : x \in \mathbb{N} \text{ and } 4 < x < 8\}$,
find:

(i) $A \cap B$ (ii) $A \cup B$ (iii) $(A \cup B) \cap C$

6. If
$A = \{2, 4, 6, 8, 10, 12\}$
and
$B = \{3, 4, 5, 6, 7, 8, 10\}$,
find:

(i) $(A – B)$ (ii) $(B – A)$ (iii) $(A – B) \cup (B – A)$

7. If
$A = \{a, b, c, d, e\}$,
$B = \{a, c, e, g\}$,
$C = \{b, e, f, g\}$,
find:

(i) $(A \cap B) – C$ (ii) $A – (B \cup C)$ (iii) $A – (B \cap C)$

8. If
$A = \{x : x \in \mathbb{N} \text{ and } x < 8\}$
and
$B = \{x : x \in \mathbb{N} \text{ and } 2x \leq 8\}$,
find:

(i) $A \cup B$ (ii) $A \cap B$ (iii) $A – B$ (iv) $B – A$

9. If $R$ is the set of all real numbers and $Q$ is the set of all rational numbers then what is the set $(R – Q)$?

10. If
$A = \{2, 3, 5, 7, 11\}$,
$B = \emptyset$,
find:

(i) $A \cup B$ (ii) $A \cap B$

11. If $A$ and $B$ are two sets such that $A \subseteq B$ then find:

(i) $A \cup B$ (ii) $A \cap B$ (iii) $A – B$

12. Which of the following sets are pairs of disjoint sets? Justify your answer:

(i) $A = \{3, 4, 5, 6\}$ and $B = \{2, 5, 7, 9\}$

(ii) $C = \{1, 2, 3, 4, 5\}$ and $D = \{6, 7, 9, 11\}$

(iii) $E = \{x : x \in \mathbb{N}, x \text{ is even and } x < 8\}$
and
$F = \{x : x = 3n, n \in \mathbb{N} \text{ and } n < 4\}$

(iv) $G = \{x : x \in \mathbb{N}, x \text{ is even}\}$ and $H = \{x : x \in \mathbb{N}, x \text{ is prime}\}$

(v) $J = \{x : x \in \mathbb{N}, x \text{ is even}\}$ and $K = \{x : x \in \mathbb{N}, x \text{ is odd}\}$

13. If
$U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$,
$A = \{1, 2, 3, 4\}$,
$B = \{2, 4, 6, 8\}$
and
$C = \{1, 4, 5, 6\}$,
find:

(i) $A’$ (ii) $B’$ (iii) $C’$ (iv) $(B’)’$

(v) $(A \cup B)’$ (vi) $(A \cap C)’$ (vii) $(B – C)’$

14. If
$U = \{a, b, c, d, e\}$,
$A = \{a, b, c\}$
and
$B = \{b, c, d, e\}$
then verify that:

(i) $(A \cup B)’ = A’ \cap B’$ (ii) $(A \cap B)’ = A’ \cup B’$

15. If $U$ is the universal set and $A \subseteq U$ then fill in the blanks:

(i) $A \cup A’ = \dots$ (ii) $A \cap A’ = \dots$ (iii) $\emptyset \cap A = \dots$ (iv) $U’ \cap A = \dots$

Answers to Practice Exercise

1.
(i) $\{a, b, c, d, e, f, g, h\}$ (ii) $\{a, c, e, g, h, m, n\}$ (iii) $\{a, b, c, d, e, f, m, n\}$

(iv) $\{e\}$ (v) $\{a, e\}$ (vi) $\{c, e\}$

2.
(i) $\{1, 2, 3, 4, 5, 6, 7, 8\}$ (ii) $\{4, 5, 6, 7, 8, 9, 10, 11\}$ (iii) $\{1, 2, 3, 4, 5, 7, 8, 9, 10, 11\}$

(iv) $\{4, 5, 6, 7, 8, 10, 11, 12, 13, 14\}$ (v) $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$ (vi) $\{7, 8\}$

(vii) $\{4, 5, 10, 11, 12, 13, 14\}$ (viii) $\{4, 5, 7, 8\}$ (ix) $\{7, 8, 9, 10, 11\}$

3.
(i) $\{7, 9\}$ (ii) $\{11\}$ (iii) $\{11, 13\}$ (iv) $\emptyset$

(v) $\{11, 13\}$ (vi) $\{7, 9, 11\}$

4.
(i) $B$ (ii) $C$ (iii) $D$ (iv) $\emptyset$

(v) $\{2\}$ (vi) $\{2\}$

5.
(i) $\{4, 6\}$ (ii) $\{2, 4, 5, 6\}$ (iii) $\{5, 6\}$

6.
(i) $\{2, 12\}$ (ii) $\{3, 5, 7\}$ (iii) $\{2, 3, 5, 7, 12\}$

7.
(i) $\{a, c\}$ (ii) $\{d\}$ (iii) $\{a, b, c, d\}$

8.
(i) $\{1, 2, 3, 4, 5, 6, 7, 8\}$ (ii) $\{2, 4, 6, 8\}$ (iii) $\{1, 3, 5, 7\}$ (iv) $\{8\}$

9. $(R – Q) = \{x : x \in \mathbb{R}, x \text{ is irrational}\}$

10.
(i) $\{2, 3, 5, 7, 11\}$ (ii) $\emptyset$

11. (i) $B$ (ii) $A$ (iii) $\emptyset$

12. (ii) $C$ and $D$, since $C \cap D = \emptyset$ (v) $J$ and $K$, since $J \cap K = \emptyset$

13.
(i) $\{5, 6, 7, 8, 9\}$ (ii) $\{1, 3, 5, 7, 9\}$ (iii) $\{2, 3, 7, 8, 9\}$ (iv) $\{2, 4, 6, 8\}$

(v) $\{5, 7, 9\}$ (vi) $\{2, 3, 5, 6, 7, 8, 9\}$ (vii) $\{1, 3, 4, 5, 6, 7, 9\}$

15.
(i) $U$ (ii) $\emptyset$ (iii) $A$ (iv) $\emptyset$

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Important Chapter Links

Set Operations is a key topic in Class 11 CBSE Mathematics that explains how different sets interact with each other. In this chapter, you will learn fundamental operations such as union, intersection, difference, complement, and symmetric difference, along with the concept of disjoint sets. These operations are essential for solving problems using Venn diagrams and understanding relationships between sets in a logical way. Important laws like De Morgan’s laws and algebra of sets are also covered to simplify complex expressions. To strengthen your understanding, this chapter includes a wide range of JEE Previous Year Questions (PYQs), MCQs, and practice problems, making it highly useful for competitive exams like JEE, NDA, and IMUCET.