1.Let $R$ be a relation on the set $\mathbb{R}$ of real numbers defined by $x R y \iff 1 + xy > 0$. Is $R$ an equivalence relation? Justify your answer with a counterexample if false.
2.Find the maximum number of equivalence relations that can be defined on a set $A = \{a, b, c\}$.
3.Let $A = \{1, 2, 3\}$. A relation $R = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)\}$. What is the minimum number of ordered pairs to be removed from $R$ so that it becomes an equivalence relation?
4.Let $S$ be the set of all sets. A relation $R$ on $S$ is defined by $A R B \iff A \subseteq B$. Prove that $R$ is a partial order relation but not an equivalence relation.
5.Let $R$ be a relation on $\mathbb{N} \times \mathbb{N}$ defined by $(a,b) R (c,d) \iff ad(b+c) = bc(a+d)$. Show that $R$ is an equivalence relation.
6.Consider the relation $R$ on $\mathbb{Z}$ given by $a R b \iff a^2 - 7ab + 6b^2 = 0$. Determine if $R$ is reflexive, symmetric, or transitive.
7.If $R_1$ and $R_2$ are two equivalence relations defined on a non-empty set $A$, prove that $R_1 \cap R_2$ is also an equivalence relation.
8.Is the union of two equivalence relations $R_1$ and $R_2$ necessarily an equivalence relation? Provide a counterexample if not.
9.Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \frac{x^2 - 1}{x^2 + 1}$. Determine whether $f$ is one-one and/or onto. Find its range.
10.Show that the function $f: \mathbb{R} \to \mathbb{R}$ given by $f(x) = x + \sin x$ is a bijection.
11.Determine whether $f(x) = e^{|x|}$ mapping from $\mathbb{R}$ to $[1, \infty)$ is bijective.
12.Let $A = \mathbb{R} - \{2\}$ and $B = \mathbb{R} - \{1\}$. If $f: A \to B$ is a function defined by $f(x) = \frac{x-1}{x-2}$, show that $f$ is a bijection and find $f^{-1}$.
13.Let $f: [0, \infty) \to [0, \infty)$ be defined by $f(x) = \frac{x}{1+x}$. Check whether $f$ is one-one and onto.
14.Consider $f: \mathbb{Z} \to \mathbb{Z}$ defined by $f(n) = \frac{n}{2}$ if $n$ is even, and $f(n) = 0$ if $n$ is odd. Find if it is one-one or onto.
15.Calculate the exact number of onto functions from a set $A$ containing 5 elements to a set $B$ containing 3 elements.
16.Find the domain of the real-valued function $f(x) = \sqrt{\log_{0.5}(x^2 - 5x + 7)}$.
17.Let $f(x) = [x]$ and $g(x) = |x|$. Find the value of $(g \circ f)(-5/3) - (f \circ g)(-5/3)$.
18.Let $f(x) = 2x - 1$ and $g(x) = x^2 + 2$ for all $x \in \mathbb{R}$. Find the solution set for the equation $(f \circ g)(x) = (g \circ f)(x)$.
19.Let $f(x) = \frac{1}{\sqrt{x - [x]}}$. Determine the domain of $f(x)$.
20.Find the range of the function $f(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}$.
21.Let $f(x) = ax+b$ and $g(x) = cx+d$. If $(f \circ g)(x) = (g \circ f)(x)$ for all real $x$, deduce the condition relating $a, b, c$, and $d$.
22.If $f(x) = \frac{x}{\sqrt{1+x^2}}$, evaluate the expression for the $n$-th composition: $(f \circ f \circ \dots \text{n times})(x)$.
23.Let $f: \mathbb{R} \to \mathbb{R}$ satisfy $f(x+y) = f(x) + f(y)$ for all $x, y \in \mathbb{R}$. Prove that $f(0) = 0$ and $f(-x) = -f(x)$.
24.Find the inverse of the function $f(x) = \log_a(x + \sqrt{x^2+1})$, where $a > 1$.
25.Determine if the function $f(x) = |x-1| + |x-2|$ from $\mathbb{R}$ to $[1, \infty)$ is bijective.
26.Find the domain of $f(x) = \sin^{-1}\left(\frac{2x-3}{5}\right) + \log_{10}(4-x^2)$.
27.Let $A$ be a set with 4 elements. How many relations on $A$ are both symmetric and reflexive?
28.If $A$ and $B$ are sets such that $n(A \times B) = 60$, $n(A) = p$ and $n(B) = q$. Find the maximum possible value of $p+q$ and minimum possible value of $p+q$.
29.Let $f(x) = \frac{x^2-3x+2}{x^2+x-6}$. Find the domain and range of $f$.
30.Solve for $x$: $2f(x) + 3f(-x) = 15 - 4x$. Hence find $f(x)$.
31.Let $R$ be a relation on $\mathbb{R}$ defined by $x R y \iff |x| \le y$. Examine if $R$ is reflexive, symmetric, and transitive.
32.If $f: \mathbb{R} \to \mathbb{R}$ is defined by $f(x) = x^3 - x$, is it one-one? Give reason.
33.Let $f(x) = 2^{x(x-1)}$ and $g(x) = x^2-1$. Find the domain of $f \circ g$.
34.Let $*$ be a binary operation on $\mathbb{Z}$ defined by $a * b = a + b - ab$. Check for commutativity and associativity.
35.Find the identity element for the binary operation $*$ on the set of rational numbers $\mathbb{Q}$, defined by $a * b = \frac{ab}{3}$.
36.Using the operation from Q35, determine the inverse of a non-zero rational number $x$.
37.Consider a binary operation $*$ on the set $S = \mathbb{R} - \{-1\}$ defined by $a * b = a + b + ab$. Prove that $*$ is commutative and associative.
38.Find the identity element for the operation defined in Q37.
39.Find the inverse of the element $2$ under the binary operation defined in Q37.
40.Show that the total number of binary operations on a set containing $n$ elements is $n^{n^2}$.
41.Find the total number of commutative binary operations on a set of 3 elements.
42.Let $A = \{1, 2, 3\}$. Consider $f = \{(1,2), (2,3), (3,1)\}$ and $g = \{(1,3), (2,1), (3,2)\}$. Show that $f \circ g = I_A$.
43.Find the inverse of $f(x) = 10^{x+1}$ where $f: \mathbb{R} \to (0, \infty)$.
44.Let $f: [1, \infty) \to [1, \infty)$ be defined by $f(x) = 2^{x(x-1)}$. Is it invertible? If so, find $f^{-1}(x)$.
45.For any real number $x$, if $f(x) = \frac{x}{1+|x|}$, prove that $f$ is strictly increasing.
46.Given $f(x) = \max(x, x^3)$ for all $x \in \mathbb{R}$. Find the values of $x$ for which $f(x) = x^3$.
47.Evaluate the number of functions $f: \{1, 2, 3\} \to \{1, 2, 3, 4, 5\}$ such that $f(1) < f(2) < f(3)$.
48.Determine whether the relation $R$ on $\mathbb{Q}$ defined by $x R y \iff x-y \in \mathbb{Z}$ is an equivalence relation.
49.Find the inverse of $f(x) = \frac{2^x - 2^{-x}}{2^x + 2^{-x}}$ defined from $\mathbb{R} \to (-1, 1)$.
50.Prove that if $f: A \to B$ is onto and $g: B \to C$ is onto, then $g \circ f$ is onto.
51.Let $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = x^2$ and $g(x) = \sin x$. Evaluate $(g \circ f)(-\sqrt{\pi/2})$.
52.A function $f: \mathbb{R} \to \mathbb{R}$ satisfies $f(x+y) = f(x) + f(y) + xy$. Find $f(x)$ if $f(1) = 1$.
53.If $f(x) = \log\left(\frac{1+x}{1-x}\right)$, show that $f(x) + f(y) = f\left(\frac{x+y}{1+xy}\right)$.
54.Check the injectivity of the function $f: (0, \infty) \to \mathbb{R}$ defined by $f(x) = x + \frac{1}{x}$.
55.Let $A = \{1, 2, 3\}$. Total number of equivalence relations having exactly 2 equivalence classes?
56.Consider the relation $R = \{(x,y): y = 2x\}$ on $\mathbb{R}$. Find $R \circ R$.
57.If $f(x) = \sqrt{a^2 - x^2}$, find $f(f(x))$. State the domain constraints clearly.
58.Let $f: \mathbb{R} \to \mathbb{R}$ be a polynomial function such that $f(x)f(1/x) = f(x) + f(1/x)$. If $f(3) = 28$, find $f(4)$.
59.Determine if $f(x) = \cos(x^2)$ is periodic. If so, find its fundamental period.
60.Evaluate the number of functions from $A = \{1,2,3,4,5\}$ to $B = \{a,b,c\}$ such that the function is onto.
