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Questions >> CBSE XII >> Math >> 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

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:

asked Dec 19, 2012 by sreemathi.v

1 answer

Let * be binary operation defined on R by $ab=1+ab, a,b \in R.$ Then the operation is

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 $\begin{array}{1 1}(i)\;a \ast b=a-b\quad a,b\in Q & (ii)\;a \ast b=a^2+b^2\quad a,b\in Q\\(iii)\;a \ast b=a+ab\quad a,b\in Q & (iv)\;a \ast b=(a-b)^2\quad a,b\in Q\end{array}$

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 $(i)\quad f\; o\; g\qquad(ii)\quad g\;o\;f\qquad(iii)\quad f\;o\;f\qquad(iv)\quad g\;o\;g$

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.

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)].

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.

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: $\; f(x)\;=\;\frac{x}{2}$

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.

asked Dec 18, 2012 by sreemathi.v

1 answer

Give an example of a map $(i)\quad which \;is\; one-one\; but\; not\; onto$

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: $\begin{array}{1 1}(a)\quad an\;injective\;mapping\;from\;A\;to\;B\\(b)\quad a\; mapping\;from\;A\;to\;B\;which\;is\;not\;injective\\(c)\quad a\;mapping\;from\;B\;to\;A \end{array}$

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.

asked Dec 18, 2012 by sreemathi.v

1 answer

If A={1,2,3,4},define relations on A which have properties of being: $\begin{array}{1 1}(a)\quad reflexive,transitive\;but\;not\;symmetric & \;\\(b)\quad symmetric\; but \;neither\;reflexive\;nor\;transitive & \;\\(c)\quad reflexive,symmetric\;and\;transitive\end{array}$

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.

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

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. $\quad f=\{(1,4),(1,5),(2,4),(3,5)\}$

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

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.

asked Dec 18, 2012 by sreemathi.v

1 answer

If the mappings f and g are given by $f\;=\;\{(1,2),(3,5),(4,1)\} \;and\;g=\{(2,3),(5,1),(1,3)\},write\;f\;o\;g$

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 $\;\{(x,y)\;:\;x \;is \;a \;person,y\; is\; the\; mother\; of\; x\}$

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 $\alpha\;and\;\beta.$

asked Dec 18, 2012 by sreemathi.v

1 answer

If $f : R \rightarrow R$ is defined by $f(x)=x^2-3x+2$ , write $f(f(x))$ .

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 $f^{-1}$ .

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 $f^{-1}.$

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 $g\;of(x)$ .

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 $D$

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.

asked Dec 18, 2012 by sreemathi.v

1 answer

Let $f:\;N\;\to N\;be\;defined\;by\;f(n)= \left\{ \begin{array}{1 1} \frac{n+1}{2}, & if\;n\;is\;odd\\ \frac{n}{2}, & if\;is\;even\end{array} \right. \qquad for\;all\;n \in N$ $\text{state whether the function f is bijective.Justify your answer.}$

asked Dec 10, 2012 by sreemathi.v

1 answer

Number of binary operations on the set $\{a, b\}$ are

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 $(0,1]$ ?.

asked Nov 27, 2012 by vaishali.a

1 answer

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

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

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?

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 $a$ .

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 $A^{-1} \;= A$ .

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.

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.

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 $\ast$ .

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 - $\;\; F=\{(a,3), (b,2), (c,1)\}$

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 $P(X)$ ?

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.

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.

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.

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.

asked Nov 21, 2012 by vaishali.a

1 answer

If $(f:R \to R)$ is defined by $f(x) = x^2$ - $3x+2$ . Find $f(f(x))$ :

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.

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 $g\;o\;f = f\;o\;g = I_R.$

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:

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.

asked Nov 21, 2012 by vaishali.a

1 answer

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