Answers posted by meena.p

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Are the following set of ordered pairs function? If so,examine whether the mapping is injective or subjective $\;\{(a,b)\;:\;a\;$ is a person,$b\;$ is an ancestor of $a\}$

answered Mar 1, 2013

Toolbox: A function $f:A \to B $ is injective. if $f(x_1)=f(x) \Rightarrow x_1=x_2$ for $x...

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Are the following set of ordered pairs function?If so,examine whether the mapping is injective or subjective $\;\{(x,y)\;:\;x \;is \;a \;person,y\; is\; the\; mother\; of\; x\}$

answered Mar 1, 2013

Toolbox: A function $f: A \rightarrow B$ where for every $x1, x2 \in X, f(x1) = f(x2) \Rightarrow...

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

answered Mar 1, 2013

Toolbox: We find values for $\alpha$ and $\beta$ such that $b=\alpha a+\beta$ for all or o...

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

answered Mar 1, 2013

Toolbox: $f:R \to R \qquad f(f(x))=fof(x)$ $ f(x)=x^2-3x+2$ $fof(x)...

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Let A={a,b,c,d} and the function f={(a,b),(b,d),(c,a),(d,c)},Write $f^{-1}$.

answered Mar 1, 2013

Toolbox: $f^{-1}$ is the inverse function of $f:R \to R$ in set A. If $fof^{-1}(x)=x$ for ...

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Let f : $R \to R$ be the function defined by $f(x)=2x-3$ $\forall x \in R$.Write $f^{-1}.$

answered Mar 1, 2013

Toolbox: $f:R \to R; $ A function $g:R \to R $ is inverse of f if $fog=I_R=gof$ when I is ...

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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 $g\;of(x)$.

answered Mar 1, 2013

Toolbox: $f;g:R \to R$ Then $gof =g(f(x)) \qquad \forall x \in R$ $f(x)=2x+1$ $g(...

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Let $D$ be the domain of the real valued function $f$ defined by $f(x)=\sqrt{25-x^2}.$Then, write $D$

answered Mar 1, 2013

Toolbox: Since f is a real valued function we find internal for n which gives values for f(x) S...

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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.

answered Mar 1, 2013

Toolbox: A relation R in a set A is called $\mathbf{ reflexive},$ if $(a,a) \in R\;$ for e...

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

answered Mar 1, 2013

The number binary operation defined by * from $\{(a,a)(a,b),(b,a),(b,b)\} \to \{a,b\}$ is $ 2^4...

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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 $(0,1]$?.

answered Feb 28, 2013

Toolbox:Given two functions $f:A \to B $ and $g:B \to C$, then composition of $f$ and $g$, $gof:A \t...

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

answered Feb 28, 2013

Toolbox: A relation R in A is an equivalence relation if it is,reflexive,symmetric and transitive...

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Let $ A = \{1, 2, 3\}$. Then number of relations containing $(1, 2)\;$ and $\;(1, 3)$ which are reflexive and symmetric but not transitive is

answered Feb 28, 2013

Toolbox:A relation R in a set A is called reflexive. if $(a,a) \in R\;for\; all\; a\in A$A relation ...

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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?

answered Feb 28, 2013

Toolbox:If $f(a)=g(a)$ for all $a \in A$ then $f$ and $g$ are equalGiven $A=\{-1,0,1,2\}, B=\{-4,-2,...

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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 $a$.

answered Feb 28, 2013

Toolbox:An element $e \in N $ is an identify element for operation * if $a*e=e*a$ for all $a \in N$T...

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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$.

answered Feb 28, 2013

Toolbox:An element $ e \in X $ is an identify element if $ e * A=A=A*e$ for $A \in X$An element A wi...

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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.

answered Feb 28, 2013

Toolbox:An operation $* : A \to B$ is commutative if $a*b=b*a$ for $a,b \in A$An operation $*:A \to ...

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Find the number of all onto functions from the set $\{1, 2, 3, ... , n\}$ to itself.

answered Feb 28, 2013

Toolbox: A function f is onto in $f:A \to A$ if for each element $x \in A$ there exists $y...

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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 $\ast$.

answered Feb 28, 2013

Toolbox:An element X is identify element for a binary operation * if $ A*X=A=X*A$An element A is inv...

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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 $P(X)$?

answered Feb 28, 2013

Toolbox: A relation R in set A is called reflexive if $(a,a) \in R$ for every $a \in A$ A rela...

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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.

answered Feb 28, 2013

Toolbox:A function $f:A \to B$ is onto if for $ y \to B$ then exists unique $ x \in A$ such that $f(...

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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.

answered Feb 27, 2013

Toolbox: A function $f: X \rightarrow Y$ where for every $x1, x2 \in X, f(x1) = f(x2) \Rig...

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Show that the function $f : R \to R$ given by $f (x) = x^3$ is injective.

answered Feb 27, 2013

Toolbox:A function $f: X \rightarrow Y$ where for every $x1, x2 \in X, f(x1) = f(x2) \Rightarrow x1 ...

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

answered Feb 27, 2013

Toolbox:Given two functions $f:A \to B $ and $g:B \to C$, then composition of $f$ and $g$, $gof:A \t...

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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.

answered Feb 27, 2013

Toolbox: To check if a function is invertible or not ,we see if the function is both one-one and...

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Let $f:R \to R$ be defined as $f(x)=10x+7.$Find the function $g:R \to R$ such that $g\;o\;f = f\;o\;g = I_R.$

answered Feb 27, 2013

Toolbox:To check if a function is invertible or not ,we see if the function is both one-one and onto...

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Consider a binary operation $\ast$ on $N$ defined as $a \ast b = a^3 + b^3$. Choose the correct answer:

answered Feb 27, 2013

Toolbox:An operation $\ast$ on $A$ is commutative if $a\ast b = b \ast a\; \forall \; a, b \in A$An ...

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Let $\ast$ be the binary operation on $N$ defined by $a \ast b=H.C.F. $ of a and b. Is $\ast$ commutative? Is $\ast$ associative? Does there exist identity for this binary operation on $N$?

answered Feb 27, 2013

Toolbox:An operation $\ast$ on $A$ is commutative if $a\ast b = b \ast a\; \forall \; a, b \in A$An ...

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Is $\ast$ defined on the set $\{1,2,3,4,5\}$ by $a\ast b=L.C.M$.of $a$ and $b$ a binary operation?

answered Feb 27, 2013

Toolbox:The lowest common multiple of two integers a and b, usually denoted by LCM(a, b), is the sma...

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Let $\ast$ be the binary operation on $N$ given by $a\ast b=L.C.M\,.of\,a\,and\,b$. Find $\begin{array}{1 1}(i)\;\; 5 \ast7,\; 20 \ast 16 & (ii)\;\; Is\; \ast \;commutative?\\(iii)\;\; Is\; \ast \;associative? & (iv)\;\; Find\, the\,identity \, of \ast \, in N\\(v)\;\; Which \, elements\, of \, N\, are\, invertible\, for\, the\, operaation\,\ast ? & \;\end{array}$

answered Feb 27, 2013

Toolbox: $a*b$ is commutative if $a_{ij}=a_{ji}$ for all $i,j \in \{1,2,3,4,5\}$ or $a*b=b...

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Consider the binary operation $ \wedge$ on the set $\{1, 2, 3, 4, 5\}$ defined by $a \wedge b = min \{a, b\}$. Write the multiplication table of the operation $ \wedge$ .

answered Feb 26, 2013

Toolbox: When number of elements in a set A is small, we can express a binary operation ∗

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For each operation $\ast$ defined below, determine whether $\ast$ is binary, commutative or associative. $\begin{array}{1 1}(i) \;\;\; On\, Z,\, define \,a*b\, = a-b & \;\\(ii) \;\;\; On\, Q,\, define \,a*b\, = ab+1 & \;\\(iii) \;\;\; On\, Q,\, define \,a*b\, = \frac {ab} {2} & \;\\(iv) \;\;\; On\, Z^+, \, define\, a*b= 2^{ab} & \;\\(v) \;\;\; On\, Z^+,\, define \,a*b\, = a^b & \;\\(vi) \;\;\; On R - \{ -1\},\, define\, a*b= \frac {a} {b+1} & \;\end{array}$

answered Feb 26, 2013

Toolbox: 1) A binary operation $*$ on set x is called commutative if $a*b=b*a$ for ever...

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For each operation $\ast$ defined below, determine whether $\ast$ is binary, commutative or associative. $\begin{array}{1 1}(i) \;\;\; On\, Z,\, define \,a*b\, = a-b & \;\\(ii) \;\;\; On\, Q,\, define \,a*b\, = ab+1 & \;\\(iii) \;\;\; On\, Q,\, define \,a*b\, = \frac {ab} {2} & \;\\(iv) \;\;\; On\, Z^+, \, define\, a*b= 2^{ab} & \;\\(v) \;\;\; On\, Z^+,\, define \,a*b\, = a^b & \;\\(vi) \;\;\; On R - \{ -1\},\, define\, a*b= \frac {a} {b+1} & \;\end{array}$

answered Feb 26, 2013

Toolbox: 1) A binary operation $*$ on set x is called commutative if $a*b=b*a$ for every $...

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For each operation $\ast$ defined below, determine whether $\ast$ is binary, commutative or associative. $\begin{array}{1 1}(i) \;\;\; On\, Z,\, define \,a*b\, = a-b & \;\\(ii) \;\;\; On\, Q,\, define \,a*b\, = ab+1 & \;\\(iii) \;\;\; On\, Q,\, define \,a*b\, = \frac {ab} {2} & \;\\(iv) \;\;\; On\, Z^+, \, define\, a*b= 2^{ab} & \;\\(v) \;\;\; On\, Z^+,\, define \,a*b\, = a^b & \;\\(vi) \;\;\; On R - \{ -1\},\, define\, a*b= \frac {a} {b+1} & \;\end{array}$

answered Feb 26, 2013

Toolbox: 1) A binary operation $*$ on set x is called commutative if $a*b=b*a$ for every $...

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For each operation $\ast$ defined below, determine whether $\ast$ is binary, commutative or associative. $\begin{array}{1 1}(i) \;\;\; On\, Z,\, define \,a*b\, = a-b & \;\\(ii) \;\;\; On\, Q,\, define \,a*b\, = ab+1 & \;\\(iii) \;\;\; On\, Q,\, define \,a*b\, = \frac {ab} {2} & \;\\(iv) \;\;\; On\, Z^+, \, define\, a*b= 2^{ab} & \;\\(v) \;\;\; On\, Z^+,\, define \,a*b\, = a^b & \;\\(vi) \;\;\; On R - \{ -1\},\, define\, a*b= \frac {a} {b+1} & \;\end{array}$

answered Feb 26, 2013

Toolbox: 1) A binary operation $*$ on set x is called commutative if $a*b=b*a$ for every $...

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For each operation $\ast$ defined below, determine whether $\ast$ is binary, commutative or associative. $\begin{array}{1 1}(i) \;\;\; On\, Z,\, define \,a*b\, = a-b & \;\\(ii) \;\;\; On\, Q,\, define \,a*b\, = ab+1 & \;\\(iii) \;\;\; On\, Q,\, define \,a*b\, = \frac {ab} {2} & \;\\(iv) \;\;\; On\, Z^+, \, define\, a*b= 2^{ab} & \;\\(v) \;\;\; On\, Z^+,\, define \,a*b\, = a^b & \;\\(vi) \;\;\; On R - \{ -1\},\, define\, a*b= \frac {a} {b+1} & \;\end{array}$

answered Feb 26, 2013

Toolbox: 1) A binary operation $*$ on set x is called commutative if $a*b=b*a$ for every $...

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Determine whether or not each of the definition of $\ast$ given below gives a binary operation. In the event that $\ast$ is not a binary operation, give justification for this - On$ \; Z^+,\,$ defined $*\,$ by$\; a*b= a-b $

answered Feb 26, 2013

Toolbox:A binary operation $∗$ on a set $A$ is a function $∗$ from $A \times A$ to $ A$. Therefore,

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Let $f : R - \{ - \frac {4} {3} \} \to R $ be a function defined as $f(x)= \frac {4x} {3x+4} $ .The inverse of $f$ is the map $g$: Range $f \to R - \{ - \frac {4} {3} \} $ given by

answered Feb 26, 2013

Toolbox:A function $g$ is called inverse of $f:x \to y$, then exists $g:y \to x$ such that $ gof=I_x...

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If $f: R\to R$ be given by $f(x)=(3-x^3)^\frac {1}{3} $, then $fof(x)$ is

answered Feb 26, 2013

Toolbox:Given two functions $f:A \to B $ and $g:B \to C$, then composition of $f$ and $g$, $gof:A \...

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Let $f : X \to Y$ be an invertible function. Show that $f$ has unique inverse.

answered Feb 26, 2013

Toolbox: To show that $f:X \to Y $ has unique inverse function we take two functions $g_1\; and \...

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Consider $f:R_+ \to [\;\text{–5}, \infty )$given by $f(x)=9x^2 +6x$-$5$.Show that $f$ is invertible with $ f^{-1} (y) = \bigg(\frac{(\sqrt{y+6}) -1} { 3}\bigg) $

answered Feb 26, 2013

Toolbox:To check if a function is invertible or not, we see if the function is both one-one and onto...

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Consider $ f:R_+ \to [4,\infty)$given by $f(x)=x^2 +4$. Show that $f$ is invertible with the inverse $f^{-1}$of $f$ given by $f^{-1}(y) = \sqrt {y-4}$,where $R_+$ is the set of all non-negative real numbers.

answered Feb 26, 2013

Toolbox:To check if a function is invertible or not ,we see if the function is both one-one and onto...

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Consider $f:R \to R$ given by $f(x)=4x+3$. Show that $f$ is invertible. Find the inverse of $ f$

answered Feb 26, 2013

Toolbox:To check if a function is invertible or not ,we see if the function is both one-one and onto...

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Show that $f : [-1,1]$ $ \rightarrow R$, given by $f(x) =\frac {x } { (x+2)}$ is one-one. Find the inverse of of the $f : [-1,1] \rightarrow $ Range $ f$.

answered Feb 25, 2013

Toolbox:To check if a function is invertible or not ,we see if the function is both one-one and onto...

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State with reason whether following functions have inverse: (iii) $h: \{2,3,4,5,\} \to \{7,9,11,13\}$ with $h=\{(2,7),(3,9),(4,11),(5,13) \} $

answered Feb 25, 2013

Toolbox:To check if a function is invertible or not ,we see if the function is both one-one and onto...

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If $f(x) = \frac { (4x+3) } { (6x-4) }, x \neq \frac {2} {3}$, show that $ f(x) =x $, for all $ x \neq \frac {2} {3}$. What is the inverse of $f$

answered Feb 25, 2013

Toolbox:A function $g$ is called inverse of $f$, if there exists $g:y \to x$ such that $g of =I_x$ a...

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Let $f,\, g\, and\, h$ be functions from $R\, to\, R.$ Show that \[ (f+g) oh = foh + goh\]

answered Feb 25, 2013

Toolbox:Given two functions $f:A \to B $ and $g:B \to C$, then composition of $f$ and $g$, $gof:A \...

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Find $ gof$ and $fog$, if (i) $ f(x) = |\;x\;| \, and \, g(x) = |\;5x-2\;| $

answered Feb 25, 2013

Toolbox: Let f,g, be two functions, the composition of functions gof is defined by, $(gof)...

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

answered Feb 24, 2013

Toolbox:A function $f: X \rightarrow Y$ where for every $x1, x2 \in X, f(x1) = f(x2) \Rightarrow x1 ...

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Show that the 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.

answered Feb 24, 2013

Toolbox: A function $f: A \rightarrow B$ where for every $x1, x2 \in X, f(x1) = f(x2) \Rig...

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