Sets

NCERT Class 11 Maths • Exercise 1.4

1. Union of Sets ($A \cup B$)

Q1: Find the union of each of the following pairs.

(i) $X = \{1, 3, 5\}, Y = \{1, 2, 3\}$

Combine all elements, no repetition.
$X \cup Y = \{1, 2, 3, 5\}$

(ii) $A = \{a, e, i, o, u\}, B = \{a, b, c\}$

$A \cup B = \{a, b, c, e, i, o, u\}$

(iii) Set Builder Form:

$A = \{x : x = 3n\} = \{3, 6, 9, …\}$ (Multiples of 3)
$B = \{x : x < 6, x \in N\} = \{1, 2, 3, 4, 5\}$
$A \cup B = \{1, 2, 4, 5, 3, 6, 9, 12, …\}$
$= \{x : x = 1, 2, 4, 5 \text{ or } x \text{ is a multiple of } 3\}$

(iv) $A = \{x : 1 < x \le 6\}, B = \{x : 6 < x < 10\}$

$A = \{2, 3, 4, 5, 6\}$
$B = \{7, 8, 9\}$
$A \cup B = \{2, 3, 4, 5, 6, 7, 8, 9\}$

(v) $A = \{1, 2, 3\}, B = \phi$

Union with empty set is the set itself.
$A \cup B = \{1, 2, 3\}$

2 & 3. Subsets and Unions

Q2: Let $A=\{a,b\}, B=\{a,b,c\}$. Is $A \subset B$? What is $A \cup B$?

Is $A \subset B$?
Yes, every element of A is in B.

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

Q3: If $A \subset B$, what is $A \cup B$?

If every element of A is already inside B, adding A to B adds nothing new.
Therefore, $A \cup B = B$.

4. Multiple Set Unions

Given: $A=\{1,2,3,4\}, B=\{3,4,5,6\}, C=\{5,6,7,8\}, D=\{7,8,9,10\}$

Q4: Find the following unions.

(i) $A \cup B$: $\{1, 2, 3, 4, 5, 6\}$
(ii) $A \cup C$: $\{1, 2, 3, 4, 5, 6, 7, 8\}$
(iii) $B \cup C$: $\{3, 4, 5, 6, 7, 8\}$
(iv) $B \cup D$: $\{3, 4, 5, 6, 7, 8, 9, 10\}$
(v) $A \cup B \cup C$: $\{1, 2, 3, 4, 5, 6, 7, 8\}$
(vi) $A \cup B \cup D$: $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$
(vii) $B \cup C \cup D$: $\{3, 4, 5, 6, 7, 8, 9, 10\}$

5 & 6. Intersection ($A \cap B$)

Q5: Find intersection of sets from Q1.

(i) $X \cap Y = \{1, 3\}$ (Common elements)
(ii) $A \cap B = \{a\}$
(iii) $\{3, 6, 9…\} \cap \{1, 2, 3, 4, 5\} = \{3\}$
(iv) $\{2, 3, 4, 5, 6\} \cap \{7, 8, 9\} = \phi$ (Disjoint)
(v) $\{1, 2, 3\} \cap \phi = \phi$

Q6: Complex Intersections.

Given: $A=\{3,5,7,9,11\}, B=\{7,9,11,13\}, C=\{11,13,15\}, D=\{15,17\}$

(i) $A \cap B$: $\{7, 9, 11\}$
(ii) $B \cap C$: $\{11, 13\}$
(iii) $A \cap C \cap D$: No element common to all three. $\to \phi$
(iv) $A \cap C$: $\{11\}$
(v) $B \cap D$: $\phi$
(vi) $A \cap (B \cup C)$:
$B \cup C = \{7, 9, 11, 13, 15\}$
$A \cap \{…\} = \{7, 9, 11\}$
(vii) $A \cap D$: $\phi$
(viii) $A \cap (B \cup D)$:
$B \cup D = \{7, 9, 11, 13, 15, 17\}$
Intersection with A = $\{7, 9, 11\}$
(ix) $(A \cap B) \cap (B \cup C)$:
$\{7, 9, 11\} \cap \{7, 9, 11, 13, 15\} = \{7, 9, 11\}$
(x) $(A \cup D) \cap (B \cup C)$:
$\{3, 5, 7, 9, 11, 15, 17\} \cap \{7, 9, 11, 13, 15\}$
$= \{7, 9, 11, 15\}$

7. Sets of Numbers

Q7: Intersections of Natural numbers.

$A = \text{Natural}, B = \text{Even}, C = \text{Odd}, D = \text{Prime}$

(i) $A \cap B$: $B$ (Even numbers are natural).
(ii) $A \cap C$: $C$ (Odd numbers are natural).
(iii) $A \cap D$: $D$ (Prime numbers are natural).
(iv) $B \cap C$: $\phi$ (No number is both even and odd).
(v) $B \cap D$: $\{2\}$ (Only even prime number).
(vi) $C \cap D$: $\{x : x \text{ is an odd prime number}\}$.

8. Disjoint Sets

[Image of Venn diagram of disjoint sets]

Q8: Which pairs are disjoint?

Definition: Sets are disjoint if $A \cap B = \phi$.

(i) $\{1, 2, 3, 4\}$ and $\{x : 4 \le x \le 6\}$
Second set is $\{4, 5, 6\}$. Common element: 4.
Result: Not Disjoint

(ii) $\{a, e, i, o, u\}$ and $\{c, d, e, f\}$
Common element: e.
Result: Not Disjoint

(iii) Even Integers and Odd Integers
No common elements.
Result: Disjoint

9, 10, 11. Difference of Sets ($A – B$)

Q9: Find Differences.

Given: $A=\{3, 6, 9, 12, 15, 18, 21\}$, $B=\{4, 8, 12, 16, 20\}$, $C=\{2, 4, 6, 8, 10, 12, 14, 16\}$, $D=\{5, 10, 15, 20\}$

(i) $A – B = \{3, 6, 9, 15, 18, 21\}$ (Removed 12)
(ii) $A – C = \{3, 9, 15, 18, 21\}$ (Removed 6, 12)
(iii) $A – D = \{3, 6, 9, 12, 18, 21\}$ (Removed 15)
(iv) $B – A = \{4, 8, 16, 20\}$
(v) $C – A = \{2, 4, 8, 10, 14, 16\}$
(vi) $D – A = \{5, 10, 20\}$
(vii) $B – C = \{20\}$
(viii) $B – D = \{4, 8, 12, 16\}$
(ix) $C – B = \{2, 6, 10, 14\}$
(x) $D – B = \{5, 10, 15\}$
(xi) $C – D = \{2, 4, 6, 8, 12, 14, 16\}$
(xii) $D – C = \{5, 15, 20\}$

Q10: X and Y differences.

$X = \{a, b, c, d\}, Y = \{f, b, d, g\}$
(i) $X – Y = \{a, c\}$
(ii) $Y – X = \{f, g\}$
(iii) $X \cap Y = \{b, d\}$

Q11: $R – Q$

$R$ = Real Numbers.
$Q$ = Rational Numbers.
Real numbers consist of Rationals and Irrationals ($T$).
Therefore, removing rationals from reals leaves irrationals.
$R – Q = T$ (Set of Irrational Numbers)

12. True/False (Disjoint Check)

Q12: State True or False.

(i) $\{2, 3, 4, 5\}$ and $\{3, 6\}$ are disjoint.
Common: 3. $\to$ False.
(ii) $\{a, e, i, o, u\}$ and $\{a, b, c, d\}$ are disjoint.
Common: a. $\to$ False.
(iii) $\{2, 6, 10, 14\}$ and $\{3, 7, 11, 15\}$ are disjoint.
Common: None. $\to$ True.
(iv) $\{2, 6, 10\}$ and $\{3, 7, 11\}$ are disjoint.
Common: None. $\to$ True.