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Recent questions in Relations and Functions
Questions
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CBSE XII
>>
Math
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Relations and Functions
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
cbse
class12
ch1
q29
p13
objective
exemplar
sec-a
math
asked
Dec 19, 2012
by
sreemathi.v
1
answer
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 where $a,b\in T$, then $R$ is:
cbse
class12
ch1
q28
p13
objective
exemplar
sec-a
math
asked
Dec 19, 2012
by
sreemathi.v
1
answer
Let * be binary operation defined on R by $ a*b=1+ab, a,b \in R.$ Then the operation * is
cbse
class12
ch1
q27
p13
exemplar
sec-b
math
asked
Dec 19, 2012
by
sreemathi.v
1
answer
Let $\ast $ be the binary operation defined on Q.Find which of the following binary operations are commutative\begin{array}{1 1}(i)\;a \ast b=a-b\quad a,b\in Q & (ii)\;a \ast b=a^2+b^2\quad a,b\in Q\\(iii)\;a \ast b=a+ab\quad a,b\in Q & (iv)\;a \ast b=(a-b)^2\quad a,b\in Q\end{array}
cbse
class12
ch1
q26
p13
exemplar
math
sec-a
asked
Dec 19, 2012
by
sreemathi.v
1
answer
Functions $ f,g:R \rightarrow R $ are defined,respectively,by $f(x)=x^2+3x+1,g(x)=2x-3,$ find\[(i)\quad f\; o\; g\qquad(ii)\quad g\;o\;f\qquad(iii)\quad f\;o\;f\qquad(iv)\quad g\;o\;g\]
cbse
class12
ch1
q25
p13
exemplar
math
asked
Dec 19, 2012
by
sreemathi.v
1
answer
Using the definition,prove that the function $f:\;A\rightarrow B$ is invertible if and only if f is both one-one and onto.
cbse
class12
ch1
q24
p13
exemplar
sec-a
math
asked
Dec 19, 2012
by
sreemathi.v
1
answer
Let A={1,2,3....9} and R be the relation in AxA defined by (a,b)R(c,d) if a+d=b+c for (a,b),(c,d) in AxA.Prove that R is an equivalence relation and also obtain the equivalent class[(2,5)].
cbse
class12
ch1
q23
p13
exemplar
sec-b
math
asked
Dec 19, 2012
by
sreemathi.v
1
answer
Each of the following defines a relation on N : \begin{array}{1 1} (i)\quad x\; is\; greater\; than\; y,x,y\quad N\\(ii)\quad x+y=10,x,y\quad N\\(iii)\quad x\;y\;is\;square\; of\; an\; integer\;x,y\quad N\\(iv)\quad x+4y=10\;x,y\quad N\end{array}Determine which of the above relations are reflexive,symmetric and transitive.
cbse
class12
ch1
q22-1
p12
exemplar
math
asked
Dec 19, 2012
by
sreemathi.v
1
answer
Let $A=[-1,1].$ Then,dicuss whether the following functions defined on $A$ are one-one,onto or bijective:$\; f(x)\;=\;\frac{x}{2} $
cbse
class12
ch1
q21-1
p12
exemplar
sec-b
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Let A=R-{3},B=R-{1}.Let $f \;:\;A \rightarrow B$ be defined by $f(x)=\Large{\frac{x-2}{x-3}}\normalsize \forall x \in A$.Then show that f is bijective.
cbse
class12
ch1
q20
p12
exemplar
sec-b
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Give an example of a map$(i)\quad which \;is\; one-one\; but\; not\; onto$
cbse
class12
ch1
q19-1
exemplar
math
sec-a
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Given A={2,3,4},B={2,5,6,7}.Construct an example of each of the following:\begin{array}{1 1}(a)\quad an\;injective\;mapping\;from\;A\;to\;B\\(b)\quad a\; mapping\;from\;A\;to\;B\;which\;is\;not\;injective\\(c)\quad a\;mapping\;from\;B\;to\;A \end{array}
cbse
class12
ch1
q18
p12
exemplar
sec-b
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Let $R$ be relation defined on the set of natural number $N$ as follows:\[R=\{(x,y):x\;\;\;N,y\;\;\;N,2x+y=41\}.\]Find the domain and range of the relation R.Also verify whether R is reflexive,symmetric and transitive.
cbse
class12
ch1
q17
p12
exemplar
sec-b
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
If A={1,2,3,4},define relations on A which have properties of being:\begin{array}{1 1}(a)\quad reflexive,transitive\;but\;not\;symmetric & \;\\(b)\quad symmetric\; but \;neither\;reflexive\;nor\;transitive & \;\\(c)\quad reflexive,symmetric\;and\;transitive\end{array}
cbse
class12
ch1
q16
p12
exemplar
sec-b
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Let $f\;:\;R\rightarrow R$ be the function defined by $f(x)=\frac{1}{2-\cos x}\forall x\;\in R.$Then,find the range of f.
cbse
class12
ch1
q14
p12
short-answer
exemplar
sec-a
easy
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
If functions $f\;:\;A \rightarrow B\;and\;g\;:\;B \rightarrow A $ satisfy g o f =$I_A$ ,then is f one-one and g onto
cbse
class12
ch1
q13
p11
short-answer
exemplar
sec-a
easy
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Let X={1,2,3} and Y={4,5}.Find whether the following subsets of X x Y are functions from X to Y or not.$\quad f=\{(1,4),(1,5),(2,4),(3,5)\} $
cbse
class12
ch1
q12
q12-1
p11
short-answer
exemplar
sec-a
easy
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Let the function $f\;:\;R \rightarrow R$ be defined by$ f(x)=\cos x,\forall \;x\;\in R$.Show that f is neither one-one nor onto
cbse
class12
ch1
q11
p11
short-answer
exemplar
sec-a
easy
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Let C be the set of complex numbers.Prove that the mapping $f\; :\;C \rightarrow R\;given\;by\;f(z)=|\;z\;|,\forall\; z \in C,$is neither one-one nor onto.
cbse
class12
ch1
q10
p11
short-answer
exemplar
sec-a
easy
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
If the mappings f and g are given by\[f\;=\;\{(1,2),(3,5),(4,1)\} \;and\;g=\{(2,3),(5,1),(1,3)\},write\;f\;o\;g\]
cbse
class12
ch1
q9
p11
short-answer
exemplar
sec-a
easy
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Are the following set of ordered pairs function?If so,examine whether the mapping is injective or subjective $\;\{(x,y)\;:\;x \;is \;a \;person,y\; is\; the\; mother\; of\; x\}$
cbse
class12
ch1
q8
q8-1
p11
short-answer
exemplar
sec-a
easy
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
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 $\alpha\;and\;\beta.$
cbse
class12
ch1
q7
p11
short-answer
exemplar
sec-a
easy
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
If $f : R \rightarrow R$ is defined by $f(x)=x^2-3x+2$, write $f(f(x))$.
cbse
class12
ch1
q6
p11
short-answer
exemplar
sec-a
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Let A={a,b,c,d} and the function f={(a,b),(b,d),(c,a),(d,c)},Write $f^{-1}$.
cbse
class12
ch1
q5
p11
short-answer
exemplar
sec-a
easy
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Let f : $R \to R$ be the function defined by $f(x)=2x-3$ $\forall x \in R$.Write $f^{-1}.$
cbse
class12
ch1
q4
p11
short-answer
exemplar
sec-b
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Let $f,g:R \rightarrow $R be defined by $f(x)=2x+1$ and $g(x)=x^2-2,\forall x \in R,$respectively. Then, find $g\;of(x)$.
cbse
class12
ch1
q3
p11
short-answer
exemplar
sec-a
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Let $D$ be the domain of the real valued function $f$ defined by $f(x)=\sqrt{25-x^2}.$Then, write $D$
cbse
class12
ch1
q2
p11
short-answer
exemplar
sec-a
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Let $A={a,b,c}$ and the relation $R$ be defined on $A$ as follows: $R= {(a,a)(b,c)(a,b)}$. Write the minimum number of ordered pairs to be ordered pairs to be added in $R$ to make $R$ reflexive and transitive.
cbse
class12
ch1
q1
p11
short-answer
exemplar
sec-a
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Let $f:\;N\;\to N\;be\;defined\;by\;f(n)= \left\{ \begin{array}{1 1} \frac{n+1}{2}, & if\;n\;is\;odd\\ \frac{n}{2}, & if\;is\;even\end{array} \right. \qquad for\;all\;n \in N$ \[\text{state whether the function f is bijective.Justify your answer.} \]
cbse
class12
bookproblem
ch1
sec2
q9
p11
medium
sec-b
math
asked
Dec 10, 2012
by
sreemathi.v
1
answer
Number of binary operations on the set $\{a, b\}$ are
cbse
class12
bookproblem
ch1
misc
q19
p31
sec-a
math
asked
Nov 27, 2012
by
vaishali.a
1
answer
Let \(f : R \to R\) be the Signum Function defined as \[ f(x) = \left \{ \begin {array} {1 1} 1, & \quad \text { x $>$ 0} \\ 0, & \quad \text { x $=$0} \\-1, & \quad \text { x $<$0} \\ \end {array} \right. \] and \(g:R \to R\) be the greatest Integer Function given by \(g(x)=[x]\) where \([x]\) is a greatest integer less thar or equal to \(x\) Then, does \(fog\) and \(gof\) coincide in \((0,1]\)?.
cbse
class12
bookproblem
ch1
misc
q18
p31
sec-b
math
asked
Nov 27, 2012
by
vaishali.a
1
answer
Let $A = \{1, 2, 3\}$. Then number of equivalence relations containing $(1, 2)$ is
cbse
class12
bookproblem
ch1
misc
q17
p30
sec-a
math
asked
Nov 27, 2012
by
vaishali.a
1
answer
Let $ A = \{1, 2, 3\}$. Then number of relations containing $(1, 2)\;$ and $\;(1, 3)$ which are reflexive and symmetric but not transitive is
cbse
class12
bookproblem
ch1
misc
q16
p30
sec-a
math
asked
Nov 27, 2012
by
vaishali.a
1
answer
Let $A=\{\text{-1,0,1,2}\}$ and $B=\{\text{-4,-2,0,2}\} and $f,g: A $\rightarrow B$ be functions defined by $f(x)=x^2-x, \;x \in A$ and $g(x)=2 |x- \frac {1} {2} | -1,\; x \in A$. Are $f$ and $g$ equal?
cbse
class12
bookproblem
ch1
misc
q15
p30
sec-b
math
asked
Nov 26, 2012
by
vaishali.a
1
answer
Define a binary operation \(\ast\) on the set \(\{0, 1, 2, 3, 4, 5\}\) as \[ a \ast b = \left\{ \begin{array} {1 1} a+b, & \quad \text{ if a$+$b $<$ 6} \\ a+b-6, & \quad \text{ if a+b $\geq$ 6} \\ \end{array} \right. \] Show that zero is the identity for this operation and each element $a\neq0$ of the set is invertible with $6-a$ being the inverse of $a$.
cbse
class12
bookproblem
ch1
misc
q14
p30
sec-b
medium
math
asked
Nov 22, 2012
by
vaishali.a
1
answer
Given a non-empty set \( X,\) let \(\ast :\; P(X)\; \times\; P(X) \to P(X) \) be defined as \(A \ast B = \; ( A-B)\; \cup \; (B-A),\; \forall A, B \in \; P(X).\). Show that the empty set \(\emptyset \) is the identity for the operation $\ast$ and all the elemnets \(A\) of \( P(X) \) are invertible with \( A^{-1} \;= A\).
cbse
class12
bookproblem
ch1
misc
q13
p30
medium
sec-b
math
asked
Nov 22, 2012
by
vaishali.a
1
answer
Consider the binary operation $\ast :\; R \times R \rightarrow R$ and $o :\; R \times R \rightarrow R$ defined as $a \ast b = | a \text{-b}|$ and \(\;a\;o\;b=a, \forall a,\;b \in R.\) Show that \(\ast\) is commutative but not associative, \(o\) is associative but not commutative. Further, show that \(\forall\; a,\; b,\; c \in R,\; a\; \ast\; (b\; o\; c) = (a \ast b) \;o\; (a \ast c)\). [If it is so, we say that the operation $\ast$ distributes over $o$]. Does $o$ distribute over? Justify your answer.
cbse
class12
bookproblem
ch1
misc
q12
p30
sec-b
medium
math
asked
Nov 22, 2012
by
vaishali.a
1
answer
Find the number of all onto functions from the set $\{1, 2, 3, ... , n\}$ to itself.
cbse
class12
bookproblem
ch1
misc
q10
p30
sec-a
math
asked
Nov 22, 2012
by
vaishali.a
1
answer
Given a non-empty set \(X\), consider the binary operation \(\ast : P(X) × P(X) \to P(X)\) given by \(A \ast B=A \cap B\; \forall A, \) \( B \;in\; P(X),\) 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 \(\ast\).
cbse
class12
bookproblem
ch1
misc
q9
p30
sec-a
math
asked
Nov 22, 2012
by
vaishali.a
1
answer
Let $S=\{a,b,c\}\;$ and$ \;T = \{1,2,3\}$. Find the inverse of the following function \(F\) from \(S\) to \(T\), if it exists - \[\;\; F=\{(a,3), (b,2), (c,1)\}\]
cbse
class12
bookproblem
ch1
misc
q11
q11-1
p30
easy
sec-a
math
asked
Nov 22, 2012
by
vaishali.a
1
answer
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 relation on $P(X)$?
cbse
class12
bookproblem
ch1
misc
q8
p29
sec-a
math
asked
Nov 22, 2012
by
vaishali.a
1
answer
Give examples of two functions \(f: N \to N\) and \(g: N \to N\) such that \(g\;o\;f \) is onto but \(f\) is not onto.
cbse
class12
bookproblem
ch1
misc
q7
p29
sec-a
math
asked
Nov 21, 2012
by
vaishali.a
1
answer
Give examples of two functions \(f : N \to Z \) and \(g: Z \to Z\) such that \(g\;o\;f\) is injective but \(g\) is not injective.
cbse
class12
bookproblem
ch1
misc
q6
p29
sec-a
math
asked
Nov 21, 2012
by
vaishali.a
1
answer
Show that the function \(f : R \to R\) given by \(f (x) = x^3\) is injective.
cbse
class12
bookproblem
ch1
misc
q5
p29
sec-a
easy
math
asked
Nov 21, 2012
by
vaishali.a
1
answer
Show that the function f:R $\rightarrow \{ x \in$ R:-1$<$x$<$1 $\}$ defined by $f(x) = \frac {x} { 1+|\;x\;|}, x \in R$ is one-one and onto function.
cbse
class12
bookproblem
ch1
misc
q4
p29
sec-a
math
asked
Nov 21, 2012
by
vaishali.a
1
answer
If $(f:R \to R)$ is defined by $f(x) = x^2$ - $3x+2$. Find $f(f(x))$:
cbse
class12
bookproblem
ch1
misc
q3
p29
sec-a
easy
math
asked
Nov 21, 2012
by
vaishali.a
1
answer
Let \(f:W \to W\) be defined as $f(n)=n$ - $1$, if \(n\;is\;odd\;and\; f(n)=n+1,\;if\;n\;is\; even.\) Show that \(f\) is invertible. Find the inverse of \(f\). Here, \(W\) is the set of all whole numbers.
cbse
class12
bookproblem
ch1
misc
q2
p29
medium
sec-b
math
modelpaper-2014
q15
asked
Nov 21, 2012
by
vaishali.a
1
answer
Let \(f:R \to R\) be defined as \(f(x)=10x+7.\)Find the function \(g:R \to R\) such that \(g\;o\;f = f\;o\;g = I_R.\)
cbse
class12
bookproblem
ch1
misc
q1
p29
sec-b
easy
modelpaper
2012
q11
math
asked
Nov 21, 2012
by
vaishali.a
1
answer
Consider a binary operation $\ast$ on $N$ defined as $a \ast b = a^3 + b^3$. Choose the correct answer:
cbse
class12
bookproblem
ch1
sec4
q13
p26
easy
sec-a
math
asked
Nov 21, 2012
by
vaishali.a
1
answer
Let $(A=N \times N \,and\, * )$ 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.
cbse
class12
bookproblem
ch1
sec4
q11
p25
sec-b
math
asked
Nov 21, 2012
by
vaishali.a
1
answer
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