Relations and Functions
Free NCERT & Exemplar step-by-step solutions — CBSE Class 12 Mathematics
NCERT Exemplar
set $A = \{ 1,2,3,4,5\}$ by $R = \left\{ {(a,b):\left| {{a^2} - {b^2}} \right| < 8} \right\}.$ Then, $R$ is given by FillBlank Q50 If $f = \{ (1,2),(3,5),(4,1)\}$ and $g = \{ (2,3),(5,1),(1,3)\} ,$ then $$gof$$ $= \ldots \ldots \ldots$ and $fog = \ldots \ldots \ldots ..$ FillBlank Q51 If $f:R \to R$ be defined by $f(x) = \frac{x}{{\sqrt {1 + {x^2}} }}$, then $({\rm{ }}fofof)(x) =$………….. FillBlank Q52 If $f(x) = \left[ {4 - {{(x - 7)}^3}} \right],$ then ${f^{ - 1}}(x) =$…………… FillBlank Q16 If $A = \{ 1,2,3,4\}$, define relations on $A$ which have properties of being
(i) reflexive, transitive but not symmetric.
(ii) symmetric but neither reflexive nor transitive.
(iii) reflexive, symmetric and transitive. LA Q17 Let $R$ be relation defined on the set of natural number $N$ as follows,
$R = \{ (x,y):x \in N,y \in N,2x + y = 41\}$.
Find the domain and range of the relation $R$.
Also verify whether $R$ is reflexive, symmetric and transitive. LA Q18 Given, $A = \{ 2,3,4\} ,B = \{ 2,5,6,7\}$.
Construct an example of each of the following
(i) an injective mapping from $A$ to $B$.
(ii) a mapping from $A$ to $B$ which is not injective.
(iii) a mapping from $B$ to $A$.
(i) which is one-one but not onto.
(ii) which is not one-one but onto.
(iii) which is neither one-one nor onto. LA Q20 Let $A = R - \{ 3\} ,B = R - \{ 1\}$.
If $f:A \to B$ be defined by $f(x) = \frac{{x - 2}}{{x - 3}}$,$\forall x \in A$.
Then, show that $f$ is bijective. LA Q21 Let $A = [ - 1,1]$, then, discuss whether the following functions defined on $A$ are one-one onto or bijective.
(i) $f(x) = \frac{x}{2}$
(ii) $g(x) = |x|$
(iii) $h(x) = x|x|$
(iv) $k(x) = {x^2}$ LA Q23 Let $A = \{ 1,2,3, \ldots ,9\}$ and $R$
be the relation in $A \times A$ defined by
$(a,b)R(c,d)$ if $a + d = b + c$ for
$(a,b),(c,d)$ in $A \times A$. Prove that $R$
is an equivalence relation and also obtain the equivalent class [(2,5)]. LA Q24 Using the definition, prove that the function $f:A \to B$ is invertible if and only if $f$ is both one-one and onto. LA Q25 Functions $f,g:R \to R$ are defined, respectively, by $f(x) = {x^2} + 3x + 1$ $g(x) = 2x - 3$, find
(i) fog
(ii) gof
(iii) fof
(iv) gog LA Q26 Let * be the binary operation defined on $Q$.
Find which of the following binary operations are commutative
(i) $a*b = a - b,\forall a,b \in Q$
(ii) $a*b = {a^2} + {b^2},\forall a,b \in Q$
(iii) $a*b = a + ab,\forall a,b \in Q$
(iv) $a*b = {(a - b)^2},\forall a,b \in Q$
LA Q27 If * be binary operation defined on $R$ by $a*b = 1 + ab,\forall a,b \in R$. Then, the operation * is(i) commutative but not associative.
(ii) associative but not commutative.
(iii) neither commutative nor associative.
(iv) both commutative and associative.
LA Q28 Let $T$ be the set of all triangles in the Euclidean plane and let a relation $R$ on $T$be defined as $aRb$, if $a$ is congruent to $b,$ $\forall a,b \in T$. Then, $R$ is MCQ Q29 Consider the non-empty set consisting of children in a family and a relation $R$ defined as $aRb$, if $a$ is brother of $b$. Then, $R$ is MCQ Q30 The maximum number of equivalence relations on the set $A = \{ 1,2,3\}$are MCQ Q31 If a relation $R$ on the set {1,2,3} be defined by $R = \{ (1,2)\}$, then $R$ is MCQ Q32 Let us define a relation $R$ in $R$ as $aRb$ if $a \ge b$. Then, $R$ is MCQ Q33 If $A = \{ 1,2,3\}$ and consider the relation
$R = \{ (1,1),(2,2),(3,3),(1,2),(2,3),(1,3)\}$
Then, $R$ is MCQ Q34 The identity element for the binary
operation * defined on $Q - \{ 0\}$ as $a*b = \frac{{ab}}{2},\forall a,b \in Q - \{ 0\}$ is MCQ Q35 If the set $A$ contains 5 elements and the set $B$ contains 6 elements, then the number of one-one and onto mappings from $A$ to $B$ is MCQ Q36 If $A = \{ 1,2,3, \ldots ,n\}$ and $B = \{ a,b\}$.
Then, the number of surjections from $A$ into $B$ is
MCQ Q37 If $f:R \to R$ be defined by $f(x) = \frac{1}{x},\forall x \in R$. Then, $f$ is MCQ Q38 If $f:R \to R$ be defined by $f(x) = 3{x^2} - 5$ and $g:R \to R$ by $g(x) = \frac{x}{{{x^2} + 1}}$.Then, gof is
MCQ Q39 Which of the following functions from $Z$ into $Z$ are bijections? MCQ Q40 If $f:R \to R$ be the functions defined by $f(x) = {x^3} + 5$, then ${f^{ - 1}}(x)$ is MCQ Q41 If $f:A \to B$ and $g:B \to C$ be the bijective functions, then ${(gof)^{ - 1}}$ is MCQ Q42 If $f:R - \left\{ {\frac{3}{5}} \right\} \to R$ be defined by $f(x) = \frac{{3x + 2}}{{5x - 3}}$, then MCQ Q43 If $f:[0,1] \to [0,1]$ be defined by $f(x) = \left\{ {\begin{array}{cccccccccccccccccccc}{x,}&{{\rm{ if \quad }}x{\rm{ \quad is \quad rational }}}\\{1 - x,}&{{\rm{ if \quad }}x{\rm{ \quad is \quad irrational }}}\end{array}} \right.$.then $(fof)x$ is
MCQ Q44 If $f:$ MCQ Q45 If $f:N \to R$ be the function defined by$f(x) = \frac{{2x - 1}}{2}$ and $g:Q \to R$ be
another function defined by $g(x) = x + 2$.
Then, $(gof)\frac{3}{2}$ is
MCQ Q46 If $f:R \to R$ be defined by $f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{2x:x > 3}\\{{x^2}:1 < x \le 3}\\{3x:x \le 1}\end{array}} \right.$.Then, $f( - 1) + f(2) + f(4)$ is
MCQ Q47 If $f:R \to R$ be given by $f(x) = \tan x$, then ${f^{ - 1}}(1)$ is MCQ Q1 Let $A = \{ a,b,c\}$ and the relation$R$ be defined on $A$ as follows
$R = \{ (a,a),(b,c),(a,b)\}$
Then, write minimum number of ordered pairs to be
added in $R$ to make $R$ reflexive and transitive.
function $f$ defined by $f(x) = \sqrt {25 - {x^2}}$. Then, write $D$.
SA Q3 If $f,g:R \to R$ be defined by $f(x) = 2x + 1$ and $g(x) = {x^2} - 2,\forall x \in R$respectively. Then, find $gof$ SA Q4 Let $f:R \to R$ be the function defined by $f(x) = 2x - 3,\forall x \in R$. Write ${f^{ - 1}}$. SA Q5 If $A = \{ a,b,c,d\}$ and the function
$f = \{ (a,b),(b,d),(c,a),(d,c)\}$, write ${f^{ - 1}}$. SA Q6 If $f:R \to R$ is defined by $f(x) = {x^2} - 3x + 2$, write $f\{ f(x)\}$ SA Q7 Is $g = \{ (1,1),(2,3),(3,5),(4,7)\}$ a function? If $g$ is described by $g(x) = \alpha x + \beta$,
then what value should be assigned to $\alpha$
and $\beta$ ? SA Q8 Are the following set of ordered pairs functions?
If so examine whether the mapping is injective or surjective.
(i) $\{ (x,y):x$ is a person, $y$ is the mother of $x\}$.
(ii) $\{ (a,b):a$ is a person, $b$ is an ancestor of $a\}$. SA Q9 If the mappings $f$ and $g$ are given by
$f = \{ (1,2),(3,5),(4,1)\}$ and $g = \{ (2,3),(5,1),(1,3)\}$, write $fog$. SA Q10 Let $C$ be the set of complex numbers.
Prove that the mapping $f:\mathcal{C} \to R$
given by $f(z) = |z|,\forall z \in C$, is
neither one-one nor onto. SA Q11 Let the function $f:R \to R$ be defined
by $f(x) = \cos x,\forall x \in R$. Show that
$f$ is neither one-one nor onto. SA Q12 Let $X = \{ 1,2,3\}$ and $Y = \{ 4,5\}$.
Find whether the following subsets of $X \times Y$ are functions from $X$ to $Y$ or not.
(i) $f = \{ (1,4),(1,5),(2,4),(3,5)\}$
(ii) $g = \{ (1,4),(2,4),(3,4)\}$
(iii) $h = \{ (1,4),(2,5),(3,5)\}$
(iv) $k = \{ (1,4),(2,5)\}$
satisfy $gof = {I_A}$, then show that $f$ is
one-one and $g$ is onto. SA Q14 Let $f:R \to R$ be the function defined
by $f(x) = \frac{1}{{2 - \cos x}},\forall x \in R$.
Then, find the range of $f$. SA Q15 Let $n$ be a fixed positive integer.
Define a relation $R$ in $Z$ as follows $\forall a$, $b \in Z,aRb$ if and only if $a - b$ is
divisible by $n$. Show that $R$ is an equivalence relation. SA
Exercise 1.1
(i) Relation R in the set. A $= \{1, 2, 3, ………, 13, 14\}$ defined as $R = \{ (x,\;y):3x - y = 0\}$
SA Q2 Show that the relation R in the set R of real numbers, defined as $A = \{ (a,\;b):a \le {b^2}\} ,$ is neither reflexive nor symmetric nor transitive. SA Q3 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. SA Q4 Show that the relation R in R defined as $R = \{ (a,\;b):a \le b\} ,$ is reflexive and transitive but not symmetric. SA Q5 Check whether the relation R in R defined by $R = \{ (a,\;b):a \le {b^3}\}$ is reflexive, symmetric or transitive. SA Q6 Show that the relation R in the set $\{1, 2, 3\}$ given by R $= \{(1, 2), (2, 1)\}$ is symmetric but neither reflexive nor transitive. SA Q7 Show that the relation R in the set A of all the books in a library of a college, given by $R = \{ (x,\;y):$ x and y have same number of pages $\}$ is an equivalence relation. SA Q8 Show that the relation R in the set A $= \{1, 2, 3, 4, 5\}$ given by $R = \{ (a,\;b):|a - b|\;is\;even\} ,$ is an equivalence relation. Show that all the elements of $\{1, 3, 5\}$ are related to each other and all the elements of $\{2, 4\}$ are related to each other. But no element of $\{1, 3, 5\}$ is related to any element of $\{2, 4\}$. SA Q9 Show that each of the relation R in the set $A = \{ x \in Z:0 \le x \le 12\} ,$ given by(i) $R = \{ (a,\;b):|a - b|$ is a multiple of 4$\}$
(ii) $R = \{ (a,\;b):a = b\}$
is an equivalence relation.
Find the set of all elements related to 1 in each case.
SA Q10 Give an example of a relation which is(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.
SA Q11 Show that the relation R in the set A of points in a plane given by R $= \{(P, Q) :$ distance of the point P rom the origin is same as the distance of the point Q from the origin $\}$, is an equivalence relation. Further, show that the set of all points related to a point $P \ne (0,\;0)$ is the circle passing through P with origin as centre. SA Q12 Show that the relation R defined in the set A of all triangles as $R = \{ ({T_1},\;{T_2}):{T_1}$ is similar to ${T_2}\}$ ,is an equivalence relation. Consider three right angle triangles ${T_1}$ , with sides 3, 4, 5, ${T_2}$ with sides 5, 12, 13 and
${T_3}$ , with sides 6, 8, 10. Which triangles among ${T_1},\;{T_2}$ and ${T_3}$ are related? SA Q13 Show that the relation R defined in the set A of all polygons as if $R = \{ ({P_1},\;{P_2}):{P_1}$ and ${P_2}$ have same number of sides $\}$, is an equivalence relation. What is the set of the elements in A related to the right angle triangle T with sides 3, 4 and 5 ? SA Q14 Let L be the set of all lines in XY-plane and R be the relation in L defined as $R = \{ ({L_1},\;{L_2}):{L_1}$ is parallel to ${L_2} \}$. Show that R is an equivalence relation. Find the set of all lines related to the line y $=$ 2x+ 4. SA Q15 Let R be the relation in the set $\{1, 2, 3, 4\}$ given by R $= \{(1, 2), (2, 2),(1, 1), (4, 4), (1, 3), (3, 3), (3, 2)\}$. Choose the correct answer.
(A) R is reflexive and symmetric but not transitive.
(B) R is reflexive and transitive but not symmetric.
(C) R is symmetric and transitive but not reflexive.
(D) R is an equivalence relation.
SA Q16 Let R be the relation in the set N given by $R = \{ (a,\;b):a = b - 2,\;b > 6\} .$ Choose the correct answer.(A) (2, 4) $\in$ R
(B) (3, 8) $\in$ R
(C) (6, 8) $\in R$
(D) (8, 7) $\in$ R
SAExercise 1.2
(i) $f:N \to N\;\;given\;\;by\;\;f(x) = {x^2}$
(ii) $f:Z \to Z\;\;given\;\,by\;\,f(x) = {x^2}$
(iii) $f:R \to R\;\,given\;\,by\;\,f(x) = {x^2}$
(iv) $f:N \to N\;\,given\;\,by\;\,f(x) = {x^3}$
(v) $f:Z \to Z\;\,given\;\,by\;\,f(x) = {x^3}$
SA Q3 Prove that the Greatest Integer Function $f:R \to R,$ given $f(x) = \;|x|,$ is neither one-one nor onto, where $[x]$ denotes the greatest integer less than or equal to x. SA Q4 Show that Modulus Function $f:R \to R,$ given by $f(x) = |x|,$ is neither one-one nor onto, where $|x|$ is x, if x is positive or 0 and $|x|\;\;is\;\; - x,$ if x is negative. SA Q5 Show that the Signum Function $f:R \to R,$ given by$f(x) = \left\{ \begin{array}{l}1,\;\;if\;\;x > 0\\0,\;\;if\;\;x = 0\\ - 1,\;\;if\;\;x < 0\end{array} \right.$
is neither one-one nor onto. SA Q6 Let A $=$ {1, 2, 3}, B $=$ {4, 5, 6, 7} and let f$=$ {(1, 4), (2, 5), (3, 6)} be a function from A to B.Show that f is one-one. SA Q7 In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) $f:R \to R,$ defned by $f(x) = 3 - 4x$
(ii) $f:R \to R,$ defined by $f(x) = 1 + {x^2}$
SA Q8 Let, A and B be sets. Show that $f:A \times B \to B \times A$ such that $f(a,\;b) = (b,\;a)$ is bijective function. SA Q9 Let $f:N \to N$ be defined by $f(n) = \left\{ \begin{array}{l}\cfrac{{n + 1}}{2},\;\;if\;\;n\;\;is\;\;odd\\\cfrac{n}{2},\;\;\;\;\;\;\;if\;\;n\;\;is\;\;even\end{array} \right.$ for all $n \in N.$ State whether the function f is bijective f is bijective. Justify your answer. SA Q10 Let $A = R - \{ 3\} \;\;and\;\;B = R - \{ 1\} .$ Consider the function $f:A \to B$ defined by $f(x) = \left( {\cfrac{{x - 2}}{{x - 3}}} \right).$ Is f one-one and onto? Justify your answer. SA Q11 Let $f:R \to R$ be defined as $f(x) = {x^4}.$ Choose the correct answer.(A) f is one-one onto
(B) f is many-one onto
(C) f is one-one but not onto
(D) f is neither one-one nor onto.
SA Q12 Let $f:R \to R$ be defined as $f(x) = 3x.$ Choose the correct answer.(A) f is one-one onto
(B) f is many-one onto
(C) f is one-one but not onto
(D) f is neither one-one nor onto.
Exercise 1.3
$(f+g)oh = foh +goh$ SA Q3 Find gof and fog, if
(i) $f(x) = \;|x|\;\;and\;\;g(x) = \;|5x - 2|$
(ii) $f(x) = 8{x^3}\;\;and\;\;g(x) = {x^{1/3}}.$
SA Q4 If $f(x) = \cfrac{{(4x + 3)}}{{(6x - 4)}},\;x \ne \cfrac{2}{3},$ show that $fof(x) = x,$ for all $x\not = \cfrac{2}{3}.$ What is the inverse of f? SA Q5 State with reason whether following functions have inverse(i) $f:\{ 1,\;2,\;3,\;4\} \to \{ 10\} \;\;with\;\;f = \{ (1,\;10),\;(2,\;10),\;(3,\;10),\;(4,\;10)\}$
(ii) $g:\{ 5,\;6,\;7,\;8\} \to \{ 1,\;2,\;3,\;4\} \;\;with\;\;g = \{ (5,\;4),\;(6,\;3),\;(7,\;4),\;(8,\;2)\}$
(iii) $h:\{ 2,\;3,\;4,\;5\} \to \{ 7,\;9,\;11,\;13\} \;\;with\;\;h = \{ (2,\;7),\;(3,\;9),\;(4,\;11),\;(5,\;13)\}$
SA Q6 Show that $f:[ - 1,\;1] \to R,$ given by $f(x) = \cfrac{x}{{(x + 2)}}$ is one-one. Find the inverse of the function $f:[ - 1,\;1] \to Range\;f.$ (Hint $:For\;\,y \in \;Range\;\,f,\;\,y = f(x) = \cfrac{x}{{x + 2}},\;\;for\;\;some\;\,x\;\;in\;[ - 1,\;\,1],$ i.e., $x = \cfrac{{2y}}{{(1 - y)}}).$
(Hint : suppose ${g_1}$ and ${g_2}$ are two inverses of f. Then for all $y \in Y,fo{g_1}(y) = {I_Y}(y) = fo{g_2}(y).$ Use one-one ness of f).
SA Q11 Consider f : $\{1, 2, 3\} \to \{a,b,c\}$ given by $f(1) = a,f(2) = b$ and $f(3) = c.$ Find ${f^{ - 1}}$ and show that if ${({f^{ - 1}})^{ - 1}} = f.$ SA Q12 Let $f:X \to Y$ be an invertible function. Show that the inverse of ${f^{ - 1}}$is $f,$ i.e., ${({f^{ - 1}})^{ - 1}} = f.$ SA Q13 If f : R$\to$ R be given by $f(x) = {(3 - {x^3})^{1/3}}$, then fof (x) is(A) ${x^{1/3}}$
(B) ${x^3}$
(C) x
(D) $(3 - {x^3})$ .
SA Q14 Let f : $R - \left\{ { - \cfrac{4}{3}} \right\} \to R$ be a function defined as $f(x) = \cfrac{{4x}}{{3x + 4}}.$ The inverse of f is the map g : Range $f \to R - \left\{ { - \cfrac{4}{3}} \right\}$ given by(A) $g(y) = \cfrac{{3y}}{{3 - 4y}}$
(B) $g(y) = \cfrac{{4y}}{{4 - 3y}}$
(C) $g(y) = \cfrac{{4y}}{{3 - 4y}}$
(D) $g(y) = \cfrac{{3y}}{{4 - 3y}}$
SAExercise 1.4
(i) Or Z+, define $*$ by a$*$ b$=$ a$-$b
(ii) On Z+, define $*$ by a$*$ b $=$ ab
(iii) On R, define $*$ by a $*$ b $= ab^2$
(iv) On Z+, define $*$ by a $*$ b $= |a - b|$
(v) On Z+, define $*$ by a $*$ b $=$ a
SA Q2 For each operation $*$defined below, determine whether $*$ is binary, commutative or associative.(i) On Z, define a $*$ b $=$ a$-$ b
(ii) On Q, define a$*$b$=$ ab + 1
(iii) On Q, define a $*$ b $=$ $\cfrac{{ab}}{2}$
(iv) On Z+, define a$*$ b$=$ 2ab
(v) On Z+, define a$*$b$=$ ab
(vi) On R$-\{ - 1\}$, define a $*$ b$=$ $\cfrac{a}{{b + 1}}$
SA Q3 Consider the binary operation $(v)$ on the set $\{1, 2, 3, 4, 5\}$ defined by $a v b = min \{a,b \}$. Write the operation table of the operation v SA Q4 Consider a binary operation $*$ on the set $\{1, 2, 3, 4, 5\}$ given by the following multiplication table.(i) Compute (2 $*$ 3) $*$ 4 and 2 $*$ (3 $*$ 4).
(ii) Is $*$ commutative ?
(iii) Compute (2 $*$ 3) $*$ (4 $*$ 5).
(Hint : use the following table)
$\begin{array}{c|ccccc} * & 1 & 2 & 3 & 4 & 5 \\ \hline 1 & 1 & 1 & 1 & 1 & 1 \\ 2 & 1 & 2 & 1 & 2 & 1 \\ 3 & 1 & 1 & 3 & 1 & 1 \\ 4 & 1 & 2 & 1 & 4 & 1 \\ 5 & 1 & 1 & 1 & 1 & 5 \end{array}$
SA Q5 Let $*$ be binary operation on the set $\{1, 2, 3, 4, 5\}$ defined by a $*$' b $=$ H.C.F. of a and b. Is the operation $*$' same as the operation $*$ defined in Q. 4 above? Justify your answer. SA Q6 Let $*$ be the binary operation on N given by a$*$ b$=$ L.C.M. of a and b. Find(i) 5 $*$ 7, 20 $*$ 16
(ii) Is $*$ commutative ?
(iii) Is $*$ associative ?
(iv) Find the identity of $*$ in N
(v) Which elements of N are invertible for the operation $*$ ?
(i) a $*$ b $=$ a$-$b
(ii) a $*$ b $= a^2 + b^2$
(iii) a $*$ b $=$ a + ab
(iv) a $*$ b $=$ (a$- b)^2$
(v) a $*$ b $= \cfrac{{ab}}{4}$
(vi) a $*$ b $= ab^2$
Find which of the binary operations are commutative and which are associative.
SA Q10 Show that none of the operations given above has identity. SA Q11 Let A$=$N × N and $*$ be the binary operation on A defined by (a, b) $*$ (c, d ) $=$ (a + c, b + d )Show that $*$ is commutative and associative. Find the identity element for $*$ on A, if any.
SA Q12 State whether the following statements are true or false. Justify.(i) For an arbitrary binary operation $*$ on a set N, a $*$ a $=$ a $\forall a \in Q$
(ii) If $*$ is a commutative binary operation on N, then a$*$ (b$*$ c)$=$ (c $*$ b)$*$ a
SA Q13 Consider a binary operation $*$ on N defined as a $*$ b $= a^3 + b^3$. Choose the correct answer.(A) Is $*$ both associative and commutative?
(B) Is $*$ commutative but not associative?
(C) Is $*$ associative but not commutative?
(D) Is $*$ neither commutative nor associative?
SAMiscellaneous Exercise
(Hint : Consider f(x) $=$ x and g(x) $= |x|$)
SA Q7 Give example of two functions f : N $\rightarrow$ N and g : N $\rightarrow$ N such that gof is onto but f is not onto.(Hint : Consider f(x) $=$ x + 1 and g(x)$=$ $\left\{ {\begin{array}{llllllllllllllllllll}{x - 1,}&{if}&{x > 1}\\{1,}&{if}&{x = 1}\end{array}} \right.$
SA Q8 Given a non empty set X, consider P(X) which is the set of all subsets of X. Define the relation R in P(X) as follows : For subsets A, B in P(X ), ARB if and only if A$\subset$B. Is R an equivalence relations on P(X) ? Justify your answer. SA Q9 Given a non-empty set X, consider the binary operation $*$: P(X) × P(X) $\rightarrow$ P(X) given by A$*$ B $=$ A$\cap$B $\forall$ A, B in P(X), where P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in $P(X)$ with respect to the operation $*$. SA Q10 Find the number of all onto functions from the set $\{1, 2, 3, ...., n\}$ to itself. SA Q11 Let S $= \{a, b, c \}$ and T $= \{1, 2, 3\}$. Find F$^{ - 1}$ of the following functions F from S to T, if it exists.(i) F $= \{(a, 3), (b, 2), (c, 1)\}$
(ii) F$= \{(a, 2), (b, 1), (c, 1)\}$
SA Q12 Consider the binary operations $*$ : R × R $\rightarrow$ R and o : R × R$\rightarrow$ R defined as a $*$ b $= |a$ - $b|$ and aob $=$ a, $\forall a,b \in R$. Show that $*$ is commutative but not associative and o is associative but not commutative. Further, show that $\forall$a, b, c $\in$R, a $*$ ( boc ) $=$ (a $*$ b) o (a $*$ c). (If it is so, we sav that the operation $*$ distributes over the operation o]. Does o distribute over $*$? Justify your answer. SA Q13 Given a non-empty set X, let$*$ : P(X) × P(X) $\rightarrow$ P(X) be defined as A $*$ B = (A$- B) \cup$(B- A), $\forall$ A, B $\in$ P(X). Show that the empty set $\phi$ is the identity for the operation $*$ and all the elements A of P(X)are invertible with ${A^{ - 1}}$ = A.
(Hint : (A $-$ $\phi$) $\cup$($\phi$ $-$A) $=$ A and (A$-$A )$\cup$(A $-$A) $=$ A $*$ A $=$ $\phi$).
SA Q14 Define a binary operation $*$ on the set {0, 1, 2, 3, 4, 5} as $a * b = \left\{ {\begin{array}{llllllllllllllllllll}{a + b,}&{if}&{a + b < 6}\\{a + b - 6,}&{if}&{a + b \ge 6}\end{array}} \right.$Show that zero is the identity forth is operation and each element$a \ne 0$ of the set is invertible with $6 - a$ being the inverse of a.
SA Q15 Let A$= \{$ - $1, 0, 1, 2\}$, B$= \{$ - $4,$ - $2, 0, 2\}$ and f, g : A $\rightarrow$ B be function defined by f(x)$= {x^2} - x,x \in A$and$g(x) = 2\left| {x - \cfrac{1}{2}} \right| - 1,x \in A.$ Are f and g equal ? Justify your answer.(Hint : One may note that two functions f : A$\rightarrow$B and g : A$\rightarrow$B such that f(a)$=$g(a) $\forall a \in A$, are called equal functions).
SA Q16 Let A $= \{1, 2, 3\}$. Then number of relations containing (1, 2) and (1, 3), which are reflexive and symmetric but not transitive is(A) 1
(B) 2
(C) 3
(D) 4
(A) 1
(B) 2
(C) 3
(D) 4
$f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{1,}&{x > 0}\\{0,}&{x = 0}\\{ - 1,}&{x < 0}\end{array}} \right.$ and g : R$\rightarrow$R be the Greatest Integer
Function given by g(x)$=$[x], where [x] is greatest integer less than or equal to x. Then, does fog and gof coincide in (0, 1] ?
(A) 10
(B) 16
(C) 20
(D) 8