Solution:
$A \setminus (B \cup C) = {2,4,6,8}$
$B \setminus (A \cup C) = {15,20,25}$
$C \setminus (B \cup A) = {7,14,21,28}$ (note: likely a typo in the original for “2δ”, should be “28”)
Thus,
$A \setminus (B \cup C) \cup B \setminus (A \cup C) \cup C \setminus (B \cup A) = {2,4,6,8,15,20,25,7,14,21,28}$
Cardinality: 11
Example (Q4):
Total number of people = 180
Number who watched Dabang $N(D) = 95$
Number who watched Avatar and RRR $N(A \cup R) = 40$
Number who watched Dabang and RRR $N(D \cup R) = 55$
Let $x$ be the number who watched all three movies.
Answer
$$
(x-10) + (10+x) + (5+x) + (55-x) + (50-x) + x + (40-x) = 180
$$$$
x + 150 = 180 \implies x = 30
$$
Number who watched only RRR and Avatar = $40 - x = 10$
3. Numerical Answer Type (NAT)
Question 5:
Suppose $f: D \longrightarrow \mathbb{R}$ is a function defined by $f(x) = \frac{\sqrt{x^2-9}}{x+3}$, where $D \subset \mathbb{Z}$. Let $A$ be the set of integers not in the domain of $f$. Find the cardinality of $A$.
Question 6:
Consider $S = {a \mid a \in \mathbb{N}, a \leq 18}$.
Let $R_1 = {(x,y) \mid x, y \in S, y = 3x}$
Let $R_2 = {(x,y) \mid x, y \in S, y = x^2}$
Find the cardinality of $R_1 \setminus (R_1 \cap R_2)$.
Question 7:
In a Zoo, there are 6 Bengal white tigers and 7 Bengal royal tigers. Out of these, 5 are males and 10 are either Bengal royal tigers or males. Find the number of female Bengal white tigers.
Solution:
Let BW = Bengal White, BR = Bengal Royal, M = Male.
$n(BR) = 7$, $n(M) = 5$, $n(BR \cup M) = 10$
Using inclusion-exclusion:
$$
n(BR \cup M) = n(BR) + n(M) - n(BR \cap M)
$$$$
10 = 7 + 5 - n(BR \cap M) \implies n(BR \cap M) = 2
$$
So, male Bengal Royal tigers = 2, male Bengal White tigers = 3
Female Bengal White tigers = 6 - 3 = 3
Answer: 3
4. Additional Examples and Incorrect Questions (Not for Marks)
Example (Q8):
Define $R = {(A,B) \mid A \text{ and } B \text{ are cousins}}$,
and $S = {(A,B) \mid A \text{ is son of } B}$.
Solution:
(Not fully solved, but shows how relations are defined for family relationships.)
Example (Q9):
Given $R(m)$ has cardinality $m = 8$, $S(n)$ has cardinality $n = 8$, so $m + n = 16$.
Example (Q10):
Define $f = {(A,B) \mid A \text{ is son of } B}$ as a function from $P$ to $Q$.
Analysis:
Option 1: Not a function if some $L \in P$ has no image.
Option 2: Function, not injective (not one-one).
Option 3: Function, onto (every element in codomain has preimage).
Option 4: Bijective (both injective and surjective).
