Answers posted by meena.p

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Let f={(1,2)(3,5),(4,1)} and g={(2,3),(5,1),(1,3)}.Then g o f =_____________ and f o g=____________.

answered Mar 6, 2013

Toolbox: $(gof)(x)=g(f(x))$ $(fog)(x)=f(g(x))$ $ gof(x)=g(f(1))$ ...

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Let the relation R be defined on the set A={1,2,3,4,5}by R={$(a,b):\mid a^2-b^2 \mid <\;8.Then\;R\;is\;given\;by$______________.

answered Mar 6, 2013

Toolbox: R is set of ordered pair satisfying $(a,b) : |a^2-b^2| < 8 \qquad a,b \i...

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State whether the function is one-one, onto or bijective.

$f : R\; \to R$ defined by $f(x)\; =\; 1+x^2$

answered Mar 6, 2013

Toolbox: A function is one-one if $f(x)=f(y) => x=y$ function is o...

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Let $\; f:R \rightarrow R$ be given as $\;f(x)=\tan x $. Then $\;f^{-1}(1)$ is

answered Mar 5, 2013

Toolbox: $f(x)=\tan x$ To find $f^{-1}(1)$ we check if for what values $(fof^{...

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Let $f:R \rightarrow R$ be defined by $ f(x)=\left \{ \begin{array}{1 1} 2x:x>3\\x^2:1<1x\leq 3\\ 3x:x\leq1\end{array} \right.$ $Then\;f(-1)+f(2)+f(4)\;is$

answered Mar 5, 2013

Toolbox: For the given function $f(x)= \left\{ \begin{array}{1 1} 2x & \quad ;x ...

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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 $(g\;o\;f)\frac{3}{2}$ is

answered Mar 5, 2013

Toolbox: $(gof)(x)=g(f(x))$ $f:N \to R$ $f(x)=\frac{2x-1}{2}$ &...

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Let $f:[2,\infty)\rightarrow R $ be the function defined by $f(x)=x^2-4x+5,$ then the range of f is

answered Mar 5, 2013

Toolbox: Range of $f:[2,\infty) \to R$ is the set of values f(x) can take for $x \in domai...

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Let $f :[0,1]\rightarrow[0,1]$ be defined by$$ f(x)= \left \{ \begin{array}{1 1}\;x,if\;x\;is\;rational & \;\\1-x,if\;x\;is\;irrational & \;\end{array} \right.$$ Then $(f o f)x$ is $$\begin{array}{1 1}(A)\;constant & (B)\;1+x\\(C)\;x & (D)\;none\;of\;these\end{array}$$

answered Mar 5, 2013

Toolbox: $fof(x)=f(f(x))$ $f:[0,1] \to [0,1]$ $f(x)= \left\{ \begin...

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Let $f:R-\{\frac{3}{5}\}\rightarrow R$ be defined by $f(x)=\Large {\frac{3x+2}{5x-3}}$ .Then

answered Mar 5, 2013

Toolbox: A function g is the inverse function $f:R -\{\frac{3}{5}\} \to R $ if $gof=...

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Let $f:A\rightarrow B\;and\;g:B\rightarrow C$ be the bijective functions.Then $(g\;o\;f)^{-1}$ is

answered Mar 5, 2013

Toolbox: A function h(x) is inverse of (gof) ie $h(x)=(gof)^{-1}$ if $[(gof)oh](x)=x...

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Let f:$R\rightarrow R$ be the functions defined by $f(x)=x^3+5.$ Then $f^{-1}(x)$ is

answered Mar 5, 2013

Toolbox: A function g is inverse of $f:R \to R $ if $fog=gof=I_R\;ie \;g=f^{-1}$ $...

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Which of the following functions from Z into Z are bijections?\begin{array}{1 1}(A)\;f(x)=x^3 & (B)\;f(x)=x+2\\(C)\;f(x)=2x+1 & (D)\;f(x)=x^2+1\end{array}

answered Mar 5, 2013

Toolbox: A function $f: Z \to Z$ is bijective if f is both one -one and onto ie $f(x...

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Let $f:R\rightarrow R $ be defined by $f(x)=3x^2-5\; and \;g:R\rightarrow R \;by \;g(x)=\Large {\frac{x}{x^2+1}}$.Then g o f is

answered Mar 5, 2013

Toolbox: $f:R \to R$ $g:R \to R$ $(gof)(x)=g(f(x))$ Given $f:R \to R$ ...

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Let $f\;:\;R\rightarrow R$ be defined by $f(x)\;=\;\frac{1}{x}\quad x\;\in\; R.$Then f is

answered Mar 5, 2013

Toolbox: 1.A function $f:R \to R$ is one-one if $f(x)=f(y)=>x=y \qquad x,y \in R$ ...

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Let A={1,2,3,....n} and B={a,b}. Then the number of surjection from A into B is \begin{array}{1 1}(A)\quad"p_2 \qquad & (B)\quad 2"-2\\ (C)\quad 2^n-1 \qquad & (D)\quad none\;of\;these\end{array}

answered Mar 5, 2013

Toolbox: The number of surjections from set A having n elements to set B having 2 elements...

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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

answered Mar 5, 2013

Toolbox: 1. A mapping from A to B is one-one if $f(a)=f(b) \qquad a,b \in A$ $=> a=b$ ...

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The identity element for the binary operation $\ast$ defined on $Q\sim \{0\}$ as $a\ast b=\large{\frac{ab}{2}} \normalsize \;a,b\in Q\sim\{0\}$ is

answered Mar 5, 2013

Toolbox:'e' is the identity element for binary operation * on Q-{0} if $ a *e=e *a=a$ $ a* b=\f...

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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:

answered Mar 5, 2013

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

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Check the injectivity and surjectivity of the following functions: $f : Z\to Z$ given by $f(x)\; = x^3$

answered Mar 5, 2013

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

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Check the injectivity and surjectivity of the following functions:(iv) $f : N\to N\; given\; by\; f(x)\; = x^3 $

answered Mar 5, 2013

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

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Check the injectivity and surjectivity of the following functions: $f : Z\to Z \;given\; by\; f(x)\; = x^2 $

answered Mar 5, 2013

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

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

answered Mar 5, 2013

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

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

answered Mar 5, 2013

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

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

answered Mar 5, 2013

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

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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

answered Mar 5, 2013

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

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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:

answered Mar 5, 2013

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

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

answered Mar 5, 2013

Toolbox: 1. A binary operation * defined on R is commutative if $a*b=b*a$ 2. A binar...

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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)].

answered Mar 5, 2013

Toolbox: 1.R is an equivalance relation if R is a) reflexive ie $(a,b) \in A \...

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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}

answered Mar 4, 2013

Toolbox: a binary operation * defined on Q in commutative if $ a*b=b*a$ Step1: ...

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Functions $ f,g:R \rightarrow R $ are defined,respectively,by $f(x)=x^2+3x+1,g(x)=2x-3,$ find\[(i)\quad f\; o\; g\qquad(ii)\quad g\;o\;f\qquad(iii)\quad f\;o\;f\qquad(iv)\quad g\;o\;g\]

answered Mar 4, 2013

Toolbox: $1.fog$ is defined by $(fog)(x)=f(g(x))$ 2.also $gof$ is defined ...

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Using the definition,prove that the function $f:\;A\rightarrow B$ is invertible if and only if f is both one-one and onto.

answered Mar 4, 2013

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

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Each of the following defines a relation on $N : (iii)\quad x\;y\;is\;square\; of\; an\; integer\;x,y\quad N$

answered Mar 4, 2013

Toolbox: 1. A function R defined on A reflexive if $(x,x) \in R\;for\;x \in A$ 2.A r...

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Each of the following defines a relation on $ N : (ii)\quad x+y=10,x,y\quad N$

answered Mar 4, 2013

Toolbox: 1. A function R defined on A reflexive if $(x,x) \in R\;for\;x \in A$ 2.A r...

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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.

answered Mar 4, 2013

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

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Let A=[-1,1].Then,dicuss whether the following functions defined on A are one-one,onto or bijective:$ (iv)\quad k(x)\;=\;x^2$

answered Mar 4, 2013

Toolbox: 1.A function f on set A is one-one if $f(x)=f(y) =>x,y \in A$ 2.A functi...

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Let A=[-1,1].Then,dicuss whether the following functions defined on A are one-one,onto or bijective$(iii)\quad h(x)\;=\;x|x| $

answered Mar 4, 2013

Toolbox: 1.A function f on set A is one-one if $f(x)=f(y) =>x,y \in A$ 2.A functi...

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Let A=[-1,1].Then,dicuss whether the following functions defined on A are one-one,onto or bijective:$(ii)\quad g(x)\;=\;|\;x\;|$

answered Mar 4, 2013

Toolbox: 1.A function f on set A is one-one ie $f(x)=f(y)=>x=y \qquad x,y \in A$ ...

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Let $A=[-1,1].$ Then,dicuss whether the following functions defined on $A$ are one-one,onto or bijective:$\; f(x)\;=\;\frac{x}{2} $

answered Mar 4, 2013

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

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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.

answered Mar 4, 2013

Toolbox: A function$f :A \to B$ is bijective if it is one-one ie $f(x)=f(y)=>x=y \qquad...

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Give an example of a map$(iii)\quad which\; is\; neither\;one-one\;nor\;onto$

answered Mar 4, 2013

Toolbox: 1. function $:A \to B$ is one-one if $f(x)=f(y) =>x=y\qquad x,y \in A$ 2...

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

answered Mar 4, 2013

Toolbox: 1. function $:A \to B$ is one-one if $f(x)=f(y) =>x=y\qquad x,y \in A$ 2...

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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.

answered Mar 3, 2013

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

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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}

answered Mar 3, 2013

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

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Let n be a fixed positive integer.Define a relation R in Z as follows: a,b E\inE Z, aRb if and only if a-b is divisible by n. Show that R is an equivalence relation.

answered Mar 3, 2013

Toolbox:A relation R is an equivalnce relation if R is reflexive,symmetric and transitiveA relation ...

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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.

answered Mar 2, 2013

Toolbox: Range of f is the possible value f(x) can take for all $ x \in R$ $\cos x$ ...

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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

answered Mar 2, 2013

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

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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)\} $

answered Mar 2, 2013

Toolbox: The given set of ordered pair subset of $x \times y$ function if same element $a ...

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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

answered Mar 2, 2013

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

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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.

answered Mar 2, 2013

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

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

answered Mar 2, 2013

Toolbox: $fog(x)=f(g(x))$ for all $Given f=\{(1,2),(3,5),(4,1)$ $g=...

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