1.If $A = \{a, b\}$ and $B = \{1, 2, 3\}$, list all elements of $A \times B$.
2.If $P = \{1, 2\}$, find $P \times P \times P$.
3.Find $x$ and $y$ if $(x/3 + 1, y - 2/3) = (5/3, 1/3)$.
4.If $A \times B = \{(p, q), (p, r), (m, q), (m, r)\}$, find sets $A$ and $B$.
5.Let $A = \{1, 2\}$ and $B = \{3, 4\}$. How many subsets does $A \times B$ have?
6.If $n(A) = 4$ and $n(B) = 5$, what is the cardinality of $B \times A$?
7.State true or false: If $A$ is an empty set, then $A \times B$ is also an empty set.
8.If $A \subseteq B$, prove using elements that $A \times C \subseteq B \times C$.
9.Find the Cartesian product of $A = \{2\}$ and $B = \emptyset$.
10.If $n(A \times B) = 15$ and $A = \{1, 2, 3\}$, what is $n(B)$?
11.Let $A = \{1, 2, 3, 4\}$. Define a relation $R$ from $A$ to $A$ by $R = \{(x, y) : y = x + 1\}$. Write $R$ in roster form.
12.For the relation $R$ in Q11, write down its domain and range.
13.Define the universal relation on a set $A = \{a, b\}$. Write it in roster form.
14.If $A = \{3, 4, 5\}$ and $B = \{1, 2\}$, find the number of possible relations from $A$ to $B$.
15.A relation $R$ on set $\mathbb{N}$ is defined by $x R y \iff x + 2y = 8$. Write $R$ as a set of ordered pairs.
16.Find the domain of the relation $R = \{(x, y) : x \in \mathbb{N}, y \in \mathbb{N}, x^2 + y = 10\}$.
17.Is the relation $R = \{(2, 3), (3, 4)\}$ a subset of $\mathbb{N} \times \mathbb{N}$? Justify.
18.Determine the inverse relation $R^{-1}$ for $R = \{(1, 2), (3, 4), (5, 6)\}$.
19.If domain of $R$ is $\{a, b, c\}$, what is the range of $R^{-1}$?
20.Let $R$ be a relation from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $R = \{(a, b): a, b \in \mathbb{Z}, a-b \text{ is an integer}\}$. Find the domain and range of $R$.
21.Check if $R = \{(1,1), (2,2), (3,3), (1,2), (2,3)\}$ on $A = \{1, 2, 3\}$ is reflexive.
22.For the relation $R$ in Q21, is it symmetric? Why or why not?
23.Check whether the relation $R$ on the set of integers $\mathbb{Z}$ defined as $R = \{(a, b) : a \le b\}$ is symmetric.
24.Is the relation $R = \{(a, b) : a = b\}$ on $\mathbb{R}$ an equivalence relation?
25.Define an anti-symmetric relation with a simple example.
26.Let $L$ be the set of all lines in a plane and $R$ be the relation "$L_1 \perp L_2$". Check if $R$ is transitive.
27.Given $A = \{1, 2, 3\}$. Construct a relation on $A$ which is symmetric but neither reflexive nor transitive.
28.Show that the relation $R$ in the set $A = \{1, 2, 3\}$ given by $R = \{(1, 2), (2, 1)\}$ is not transitive.
29.If $R$ is an equivalence relation on set $X$, what are the properties of equivalence classes formed by $R$?
30.Find the equivalence class $[1]$ for the relation $a R b \implies a - b$ is divisible by $3$ on the set of integers $\mathbb{Z}$.
31.Examine if the relation $R = \{(2, 1), (3, 1), (4, 2)\}$ is a function. Give reasons.
32.Show that the function $f: \mathbb{N} \to \mathbb{N}$ given by $f(x) = 2x$ is one-one but not onto.
33.Is the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^2$ one-one? Justify.
34.Show that $f(x) = x^3$ mapping from $\mathbb{R} \to \mathbb{R}$ is a bijection.
35.Let $A = \{1, 2, 3\}$ and $B = \{4, 5, 6, 7\}$. If $f = \{(1, 4), (2, 5), (3, 6)\}$, verify if $f$ is one-one. Is it onto?
36.Give an example of a function which is many-one and onto.
37.Find the total number of functions from a set containing 3 elements to a set containing 2 elements.
38.Calculate the number of one-one functions from a set $A$ of 3 elements to a set $B$ of 4 elements.
39.Prove that the greatest integer function $f: \mathbb{R} \to \mathbb{R}$ given by $f(x) = [x]$ is neither one-one nor onto.
40.Check injectivity for $f: \mathbb{Z} \to \mathbb{Z}$ given by $f(x) = x^2 + 1$.
41.If $f(x) = \sin x$ and $g(x) = x^2$, find $(f \circ g)(x)$ and $(g \circ f)(x)$.
42.Given $f(x) = 2x+3$ and $g(x) = x^2+1$. Evaluate $f(g(2))$.
43.If $f(x) = \frac{x-1}{x+1}$, find $f(f(x))$.
44.State the condition for a function $f: A \to B$ to have an inverse function $f^{-1}$.
45.Find the inverse of the function $f(x) = 3x - 4$, mapping from $\mathbb{R} \to \mathbb{R}$.
46.Let $f(x) = \frac{2x+3}{x-1}, x \neq 1$. Find $f^{-1}(x)$.
47.If $f(x) = x^3$, prove that $f^{-1}(x) = x^{1/3}$.
48.Show that if $f: A \to B$ and $g: B \to C$ are one-one, then $g \circ f: A \to C$ is also one-one.
49.Let $f = \{(1, 2), (2, 3), (3, 4)\}$ and $g = \{(2, 3), (3, 1), (4, 2)\}$. Find $g \circ f$.
50.True or False: $(f \circ g)^{-1} = f^{-1} \circ g^{-1}$. Correct it if false.
51.Check whether the operation $*$ on $\mathbb{Z}$ defined by $a * b = a - b$ is commutative.
52.Determine if $*$ defined on $\mathbb{Q}$ by $a * b = ab/2$ is associative.
53.Find the identity element for the binary operation $*$ on $\mathbb{R}$ defined by $a * b = a + b - 5$.
54.For the operation in Q53, find the inverse of an element '$a$'.
55.Let $*$ be an operation on $\mathbb{N}$ given by $a * b = \text{LCM}(a, b)$. Find $20 * 16$.
56.Is the operation $a * b = \text{HCF}(a, b)$ on $\mathbb{N}$ commutative? Justify.
57.Draw the operation table for the binary operation $\lor$ (maximum) on the set $\{1, 2, 3\}$.
58.Evaluate $4 +_6 5$, where $+_6$ denotes addition modulo 6.
59.Evaluate $3 \times_5 4$, where $\times_5$ denotes multiplication modulo 5.
60.If an operation $*$ on $\mathbb{Z}^+$ is defined as $a * b = 2^{ab}$, check if it is commutative.
