1.If $A = \{1, 2, 3\}$, $B = \{3, 4\}$, and $C = \{4, 5, 6\}$, find $A \times (B \cap C)$.
2.Let $A$ and $B$ be two sets such that $n(A) = 3$ and $n(B) = 2$. If $(x, 1), (y, 2), (z, 1)$ are in $A \times B$, find $A$ and $B$, where $x, y, z$ are distinct elements.
3.The Cartesian product $A \times A$ has 9 elements among which are found $(-1, 0)$ and $(0, 1)$. Find the set $A$ and the remaining elements of $A \times A$.
4.If $A \times B \subseteq C \times D$ and $A \times B \neq \emptyset$, prove that $A \subseteq C$ and $B \subseteq D$.
5.Find the values of $x$ and $y$ if $(x^2 - 3x, y^2 - 5y) = (-2, -6)$.
6.Given $A = \{x \in \mathbb{N} : x < 3\}$ and $B = \{x \in \mathbb{W} : x < 2\}$. Find the elements of $(A \cup B) \times (A \cap B)$.
7.If $n(A \times B) = 24$ and $n(A) = 3$, find the maximum number of subsets that can be formed from set $B$.
8.Prove algebraically that $A \times (B \cup C) = (A \times B) \cup (A \times C)$.
9.If $A = \{1, 2\}$, write down the complete set for $A \times A \times A$.
10.Determine sets $A$ and $B$ if $A \times B = \{(x, a), (x, b), (y, a), (y, b), (z, a), (z, b)\}$.
11.Let $R$ be a relation on the set of natural numbers $\mathbb{N}$ defined by $x + 3y = 12$. Find the domain and range of $R$.
12.If $R$ is a relation from $A=\{11, 12, 13\}$ to $B=\{8, 10, 12\}$ defined by $y = x - 3$, write $R^{-1}$ in roster form.
13.Let $A = \{1, 2, 3, \dots, 14\}$. Define a relation $R$ from $A$ to $A$ by $R = \{(x, y) : 3x - y = 0\}$. Write down its domain, co-domain, and range.
14.Find the total number of non-empty relations that can be defined from set $A = \{a, b, c\}$ to set $B = \{1, 2\}$.
15.Let $R = \{(x, x^3) : x \text{ is a prime number less than 10}\}$. Find $R$ in roster form.
16.Define a relation $R$ on the set $\mathbb{Z}$ of integers as follows: $(x, y) \in R \iff x^2 + y^2 = 25$. Find the domain of $R$.
17.Find the range of the relation $R$ defined in Question 16.
18.Let $R_1 = \{(1, 2), (2, 3)\}$ and $R_2 = \{(2, 2), (3, 4)\}$. Find $R_1 \cup R_2$ and $R_1 \cap R_2$.
19.Let $R$ be the relation on $\mathbb{R}$ defined by $R = \{(a, b) : a \le b^2\}$. Is $(1/2, 1/2) \in R$? Justify your answer.
20.Write the domain of the relation $R$ defined by $R = \{(x, y) : |x| + |y| \le 3\}$ where $x, y \in \mathbb{Z}$.
21.Show that the relation $R$ in the set $\mathbb{Z}$ given by $R = \{(a, b) : 2 \text{ divides } (a - b)\}$ is an equivalence relation.
22.Show that the relation $R$ in the set $A = \{1, 2, 3, 4, 5\}$ given by $R = \{(a, b) : |a - b| \text{ is even}\}$ is an equivalence relation.
23.Find all elements related to $1$ in the equivalence relation defined in Question 22.
24.Prove that the relation $R$ on the set $\mathbb{R}$ of real numbers, defined as $R = \{(a, b) : a \le b^3\}$ is neither reflexive, nor symmetric, nor transitive.
25.Let $R$ be a relation on the set of all lines in a plane defined by $L_1 R L_2 \iff L_1 \parallel L_2$. Show that $R$ is an equivalence relation.
26.Show that the relation $R$ on the set $A = \{x \in \mathbb{Z} : 0 \le x \le 12\}$, given by $R = \{(a, b) : |a - b| \text{ is a multiple of } 4\}$ is an equivalence relation.
27.Write the equivalence class $[1]$ for the relation in Question 26.
28.Let $S$ be a relation on the set $\mathbb{N} \times \mathbb{N}$ defined by $(a, b) S (c, d) \iff a + d = b + c$. Prove that $S$ is an equivalence relation.
29.Check whether the relation $R$ defined in the set $\{1, 2, 3, 4, 5, 6\}$ as $R = \{(a, b) : b = a + 1\}$ is reflexive, symmetric, or transitive.
30.Give an example of a relation on a set $A=\{1, 2, 3\}$ which is symmetric and transitive but not reflexive.
31.Show that the function $f: \mathbb{R} \to \mathbb{R}$, defined as $f(x) = x^2$, is neither one-one nor onto.
32.Let $A = \mathbb{R} - \{3\}$ and $B = \mathbb{R} - \{1\}$. Consider the function $f: A \to B$ defined by $f(x) = \frac{x-2}{x-3}$. Is $f$ one-one and onto? Justify.
33.Show that the Signum Function $f: \mathbb{R} \to \mathbb{R}$, given by $f(x) = 1$ if $x>0$, $0$ if $x=0$, $-1$ if $x<0$, is neither one-one nor onto.
34.Check the injectivity and surjectivity of the function $f: \mathbb{Z} \to \mathbb{Z}$ given by $f(x) = x^3$.
35.Prove that the function $f: \mathbb{N} \to \mathbb{N}$ defined by $f(n) = n+1$ if $n$ is odd, and $f(n) = n-1$ if $n$ is even, is a bijection.
36.Let $f: [0, \pi/2] \to \mathbb{R}$ be given by $f(x) = \sin x$ and $g: [0, \pi/2] \to \mathbb{R}$ be given by $g(x) = \cos x$. Show that $f$ and $g$ are one-one, but $f+g$ is not one-one.
37.Find the number of onto functions from a set $A$ containing 4 elements to a set $B$ containing 3 elements.
38.Let $A = \{1, 2, 3\}$. Find the total number of bijective functions from $A$ to itself.
39.Show that $f: \mathbb{N} \to \mathbb{N}$ given by $f(1) = f(2) = 1$ and $f(x) = x-1$ for every $x > 2$ is onto but not one-one.
40.Determine whether $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 3 - 4x$ is a bijection.
41.Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = 3x^2 - 5$ and $g: \mathbb{R} \to \mathbb{R}$ by $g(x) = \frac{x}{x^2+1}$. Find the expression for $(f \circ g)(x)$.
42.Let $f(x) = \frac{4x+3}{6x-4}, x \neq 2/3$. Show that $(f \circ f)(x) = x$ for all $x \neq 2/3$. Deduce the inverse of $f$.
43.If $f(x) = e^x$ and $g(x) = \log_e x$ ($x>0$), find $(f \circ g)(x)$ and $(g \circ f)(x)$. Are they equal?
44.Find the inverse of the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \frac{2x-7}{4}$.
45.Consider $f: \mathbb{R}_+ \to [-5, \infty)$ given by $f(x) = 9x^2 + 6x - 5$. Show that $f$ is invertible and $f^{-1}(y) = \frac{\sqrt{y+6}-1}{3}$.
46.If $f(x) = \frac{x}{\sqrt{1+x^2}}$, prove algebraically that $(f \circ f \circ f)(x) = \frac{x}{\sqrt{1+3x^2}}$.
47.Let $f, g: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = |x|$ and $g(x) = |x-1|$. Find the value of $(g \circ f)(-2)$.
48.Are $f \circ g$ and $g \circ f$ always defined? Explain with a scenario where one is defined and the other is not.
49.Let $S = \{1, 2, 3\}$. Determine whether the function $f: S \to S$ defined as $f = \{(1,2), (2,1), (3,1)\}$ has an inverse. Justify.
50.If $f(x) = x+7$ and $g(x) = x-7, x \in \mathbb{R}$, find the value of $(f \circ g)(7)$.
51.Determine whether the operation $*$ on $\mathbb{Q}$ defined by $a * b = \frac{ab}{4}$ is commutative and associative.
52.Find the identity element for the binary operation $*$ defined on $\mathbb{Q}$ by $a * b = \frac{ab}{4}$.
53.For the operation in Q52, find the inverse of an element $a \in \mathbb{Q}$, where $a \neq 0$.
54.Let $*$ be a binary operation on the set $\mathbb{Q} - \{1\}$ defined by $a * b = a + b - ab$. Find the identity element.
55.Find the invertible elements for the operation defined in Q54, and write the expression for the inverse of $a$.
56.Let $*$ be a binary operation on $\mathbb{N}$ given by $a * b = \text{HCF}(a, b)$. Does there exist an identity element for this operation on $\mathbb{N}$?
57.Show that there is no identity element for the operation $a * b = a - b$ defined on $\mathbb{R}$.
58.Consider the binary operation $\land$ on the set $\{1, 2, 3, 4, 5\}$ defined by $a \land b = \min(a, b)$. Write the operation table.
59.Define a binary operation $+_6$ on the set $S = \{0, 1, 2, 3, 4, 5\}$ as $a +_6 b = (a+b) \pmod 6$. Find the inverse of the element 4.
60.Let $A = \mathbb{N} \times \mathbb{N}$ and $*$ be the binary operation on $A$ defined by $(a,b) * (c,d) = (a+c, b+d)$. Show that $*$ is commutative.
