Relation and Function Set-1

Q1: Let $T$ be the set of all triangles in the Euclidean plane, and let a relation $R$ on $T$ be defined as $a R b$ if $a$ is congruent to $b$, $a, b \in T$. Then $R$ is

Q2: The maximum number of equivalence relations on the set $A = \{2, 3, 4\}$ are

Q3: If $f : R \rightarrow R$ be the function defined by $f(x) = x^3 + 5$, then $f^{–1}(x)$ is

Q4: If a relation $R$ on the set $\{1, 2, 3\}$ be defined by $R = \{(1, 2)\}$, then $R$ is

Q5: If $f : R \rightarrow R$ be defined by $f(x) = 2/x$, $x \forall R$, then $f$ is

Q6: If $f : A \rightarrow B$ and $g : B \rightarrow C$ be the bijective functions, then $(gof)^{–1}$ is

Q7: Which of the following functions form $Z$ into $Z$ bijections?

Q8: If the set $A$ contains 7 elements and the set $B$ contains 8 elements, then number of one-one and onto mappings from $A$ to $B$ is

Q9: If $f : R – \{3/5\} \rightarrow R$ be defined by $f (x) = \frac{3x + 2}{5x - 3}$ then

Q10: Let $A = \{1, 2, 3, 4\}$. Let $R$ be the equivalence relation on $A \times A$ defined by $(a, b) R (c, d)$ if $a + d = b + c$. Then the equivalence class $[(1, 3)]$ is

Q11: Let $f : N \rightarrow R$ be the function defined by $f (x) = \frac{2x - 1}{2}$ and $g : Q \rightarrow R$ be another function defined by $g (x) = x + 2$. Then $(gof) \frac{3}{2}$ is

Q12: If $f: R\rightarrow R$ given by $f(x) =(3 − x^3)^{1/3}$, find $fof(x)$

Q13: Let $A = \{1,2,3\}$. The number of equivalence relations containing $(1,2)$ is

Q14: Let $f:R\rightarrow R$ defined by $f(x) = x^4$. Choose the correct answer

Q15: Let $f:R\rightarrow R$ defined by $f(x) = 3x$. Choose the correct answer

Q16: If $A = \{1,2,3\}$, $B = \{4,6,9\}$ and $R$ is a relation from $A$ to $B$ defined by ‘$x$ is smaller than $y$’. The range of $R$ is

Q17: The relation $R = \{ (1,1),(2,2),(3,3)\}$ on $\{1,2,3\}$ is

Q18: Let $A =\{1,2,3\}$ and consider the relation $R= \{(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)\}$ then $R$ is

Q19: Let us define a relation $R$ in $R$ as $a R b$ if $a\geq b$. Then $R$ is

Q20: Domain of $f$ is (Context: $f(x) = \frac{x − 1}{x – 2}$)

Q21: If $g : R – \{2\} \rightarrow R – \{1\}$ is defined by $g(x) = 2f(x) – 1$, then $g(x)$ in terms of $x$ is (Context: $f(x) = \frac{x − 1}{x – 2}$)

Q22: A function $f(x)$ is said to be one-one if

Q23: Range of $f$ is (Context: $f(x) = \frac{x − 1}{x – 2}$)

Q24: The function $f : A \rightarrow B$ defined by $f(x) = 4x + 7, x \in R$ is

Q25: The smallest integer function $f(x) = [x]$ is

Q26: The function $f : R \rightarrow R$ defined by $f(x) = 3 – 4x$ is

Q27: The number of bijective functions from set $A$ to itself when $A$ contains $10^6$ elements is

Q28: If $f(x) = (ax^2 + b)^3$, then the function $g$ such that $f(g(x)) = g(f(x))$ is given by

Q29: If $f : R \rightarrow R$, $g : R \rightarrow R$ and $h : R \rightarrow R$ is such that $f(x) = x^2$, $g(x) = \tan x$ and $h(x) = \log x$, then the value of $[ho(gof)](x)$, if $x = \frac{\sqrt{\pi}}{2}$ will be

Q30: If $f : R \rightarrow R$ and $g : R \rightarrow R$ defined by $f(x) = 2x + 3$ and $g(x) = x^2 + 7$, then the value of $x$ for which $f(g(x)) = 25$ is

Q31: If $f(x)=\frac{x-1}{x+1}$, then $f(f(x))$ is

Q32: If $f(x) = 1-\frac{1}{x}$, then $f(f(\frac{1}{x}))$

Q33: If $f : R \rightarrow R, g : R \rightarrow R$ and $h : R \rightarrow R$ are such that $f(x) = x^2, g(x) = \tan x$ and $h(x) = \log x$, then the value of $(go(foh)) (x)$, if $x = 1$ will be

Q34: If $f(x) = \frac{3 x+2}{5 x-3}$ then $(fof)(x)$ is

Q35: If the binary operation $*$ is defind on the set $Q^+$ of all positive rational numbers by $a * b = \frac{a b}{4}$. Then, $3 *(\frac{1}{5} * \frac{1}{2})$ is equal to

Q36: The number of binary operations that can be defined on a set of 2 elements is

Q37: Let $*$ be a binary operation on $Q$, defined by $a * b = \frac{3 a b}{5}$ is

Q38: Let $*$ be a binary operation on set $Q$ of rational numbers defined as $a * b = \frac{a b}{5}$. Write the identity for $*$.

Q39: For binary operation $*$ defind on $R – \{1\}$ such that $a * b = \frac{a}{b+1}$ is

Q40: The binary operation $*$ defind on set $R$, given by $a * b = \frac{a+b}{2}$ for all $a,b \in R$ is

Q41: Let $A = N \times N$ and $*$ be the binary operation on $A$ defined by $(a, b) * (c, d) = (a + c, b + d)$. Then $*$ is

Q42: Find the identity element in the set $I^+$ of all positive integers defined by $a * b = a + b$ for all $a, b \in I^+$.

Q43: Let $*$ be a binary operation on set $Q – \{1\}$ defind by $a * b = a + b – ab : a, b \in Q – \{1\}$. Then $*$ is

Q44: The binary operation $*$ defined on $N$ by $a * b = a + b + ab$ for all $a, b \in N$ is

Q45: The number of commutative binary operation that can be defined on a set of 2 elements is

Q46: Let $T$ be the set of all triangles in the Euclidean plane, and let a relation $R$ on $T$ be defined as $a R b$ if $a$ is congruent to $b \forall a, b \in T$. Then $R$ is

Q47: The maximum number of equivalence relations on the set $A = \{1, 2, 3\}$ are

Q48: Let us define a relation $R$ in $R$ as $a R b$ if $a \geq b$. Then $R$ is

Q49: Let $A = \{1, 2, 3\}$ and consider the relation $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)\}$. Then $R$ is

Q50: The identity element for the binary operation $*$ defined on $Q – \{0\}$ as $a * b = \frac{a b}{2}$ $\forall a, b \in Q – \{0\}$ is

Q51: Let $A = \{1, 2, 3, \ldots n\}$ and $B = \{a, b\}$. Then the number of surjections from $A$ into $B$ is

Q52: Let $f : R \rightarrow R$ be defind by $f(x) = \frac{1}{x}$ $\forall x \in R$. Then $f$ is

Q53: If $f : R \rightarrow R$ be given by $f(x) = \tan x$. Then $f^{-1}(1)$ is

Q54: Let $R$ be a relation on the set $N$ of natural numbers denoted by $nRm \Leftrightarrow n$ is a factor of $m$ (i.e. $n | m$). Then, $R$ is

Q55: Let $S = \{1, 2, 3, 4, 5\}$ and let $A = S \times S$. Define the relation $R$ on $A$ as follows: $(a, b) R (c, d)$ iff $ad = cb$. Then, $R$ is

Q56: Total number of equivalence relations defined in the set $S = \{a, b, c\}$ is

Q57: The relation $R$ is defined on the set of natural numbers as $\{(a, b) : a = 2b\}$. Then, $R^{-1}$ is given by

Q58: Let $X = \{-1, 0, 1\}$, $Y = \{0, 2\}$ and a function $f : X \rightarrow Y$ defiend by $y = 2x^4$, is

Q59: Let $A = R – \{3\}$, $B = R – \{1\}$. Let $f : A \rightarrow B$ be defined by $f(x)=\frac{x-2}{x-3}$. Then,

Q60: The mapping $f : N \rightarrow N$ is given by $f(n) = 1 + n^2, n \in N$ when $N$ is the set of natural numbers is