1.
Answer: $A \times B = \{(1,a), (1,b), (2,a), (2,b)\}$
2.
Answer: $3 \times 4 = 12$
3.
Answer: False (Unless $A = B$ or one is an empty set).
4.
Answer: $x + 1 = 3 \implies x = 2$ and $y - 2 = 1 \implies y = 3$.
5.
Answer: $\{(2,1), (3,1), (3,2)\}$
6.
Answer: A relation $R$ is a subset of the Cartesian product $A \times B$.
7.
Answer: Domain = $\{1, 2, 3\}$; Range = $\{2, 3, 4\}$.
8.
Answer: $n(A \times B) = 3 \times 2 = 6$. Total relations = $2^6 = 64$.
9.
Answer: Yes (It is called the Empty or Void Relation).
10.
Answer: The Universal Relation ($R = A \times A$).
11.
Answer: $I_A = \{(1,1), (2,2), (3,3)\}$
13.
Answer: Yes, because $(1,2) \in R \implies (2,1) \in R$ and vice versa.
14.
Answer: Yes, because $(1,1), (2,2), \text{ and } (3,3)$ are all present in $R$.
15.
Answer: False (An empty relation on a non-empty set is symmetric and transitive vacuously, but not reflexive).
16.
Answer: Reflexive, Symmetric, and Transitive.
17.
Answer: Yes, because $a - a = 0$, which is an even number.
18.
Answer: Yes, if $a - b$ is even, then $-(a - b) = b - a$ is also even (Symmetric property).
19.
Answer: An Equivalence Class.
20.
Answer: $[0] = \{\dots, -4, -2, 0, 2, 4, \dots\}$ (The set of all even integers).
21.
Answer: Every element of set $A$ must have one and only one distinct image in set $B$.
23.
Answer: To determine if a function is one-one (injective). If any horizontal line intersects the graph more than once, it is many-one.
25.
Answer: Onto function (Surjective mapping).
26.
Answer: Bijective function (or Bijection).
27.
Answer: Many-one (e.g., $f(-2) = f(2) = 4$).
28.
Answer: One-one (Linear functions without bounds are injective).
29.
Answer: No, the range is $[-1, 1]$, which is not equal to the co-domain $\mathbb{R}$.
30.
Answer: An Into function.
34.
Answer: $f(g(x)) = f(2x) = (2x)^2 = 4x^2$
35.
Answer: $g(f(x)) = g(x^2) = 2x^2$
36.
Answer: False (as proven by answers 34 and 35).
37.
Answer: Set $A$ to set $C$.
38.
Answer: $f(x+1) = (x+1) + 1 = x + 2$
43.
Answer: $A \times A$ to set $A$.
46.
Answer: $2 + 3 + 1 = 6$
47.
Answer: Yes, because $a+b+1 = b+a+1$.
48.
Answer: $3 + 4 = 7$. Remainder when $7$ is divided by $5$ is $2$. Answer is $2$.
49.
Answer: $4 \times 5 = 20$. Remainder when $20$ is divided by $6$ is $2$. Answer is $2$.
50.
Answer: $\text{max}(5, 7) = 7$.
