Recent questions tagged ch1

Let $f:\{1,3,4\} \to \{1,2,5\} $ and $g:\{1,2,5\}\to\{1,3\}$ be given by $f = \{(1, 2), (3, 5), (4, 1)\}$ and $ g = \{(1, 3), (2, 3), (5, 1)\}$. Write down $gof$.

asked Nov 9, 2012 by vaishali.a

1 answer

Let $f : R \to R$ be defined as $f (x) = 3x$. Choose the correct answer:

asked Nov 9, 2012 by vaishali.a

1 answer

Let $f: R \to R $, be defined as $f(x) = x^4$. Choose the correct answer.

asked Nov 9, 2012 by vaishali.a

1 answer

Let $A=R-\{3\}$ and $B=R-\{1\}$. Consider the function $ f:A \to B$ defined by $f(x)=\large\frac {x-2}{ x-3}$. Is $f$ one-one and onto?

asked Nov 9, 2012 by vaishali.a

1 answer

Let $A$ and $B$be sets. Show that$f : A × B \to\; B×A$such that $ f(a,b)\,=\,(b,a)$ is bijective function

asked Nov 8, 2012 by vaishali.a

1 answer

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.(i) $f : R\; \to R\; defined \; by \; f(x)\; =\; 3-4x$

asked Nov 8, 2012 by vaishali.a

1 answer

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.

asked Nov 8, 2012 by vaishali.a

1 answer

Show that the Signum Function $f : R \to R$, given by \[ f(x) = \left\{ \begin{array}{l l} 1, & if \; x \; > 0 \\ 0, & if\; x\;= 0 \text{ is neither one-one nor onto } \\ -1, & if\; x \;< 0 \end{array} \right. \]

asked Nov 8, 2012 by vaishali.a

1 answer

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.

asked Nov 8, 2012 by vaishali.a

1 answer

Prove that the Greatest Integer Function $f : R \to R$, given by $f (x) = [x]$, is neither one-one nor onto, where $[x]$ denotes the greatest integer less than or equal to $x$.

asked Nov 8, 2012 by vaishali.a

1 answer

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

asked Nov 8, 2012 by vaishali.a

1 answer

Show that the function $f : R_* \rightarrow R_* $ defined by $ f(x) = (\frac{1} {x})$ is one-one and onto, where $R_* $ is the set of all non-zero real numbers. Is the result true, if the domain $R_* $ is replaced by $N$ with co-domain being same as $R_*$?

asked Nov 8, 2012 by vaishali.a

1 answer

Let $R$ be the relation in the set $N$ given by $R=\{(a,b): a=b-2, b>6\}$. Choose the correct answer:

asked Nov 8, 2012 by vaishali.a

1 answer

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.

asked Nov 8, 2012 by vaishali.a

1 answer

Let $L$ be the set of all lines in $XY$ plane and $R$ be the relation in $L$ defined as $R = \{(L1, L2) : L1\:$ is parallel to $L2\}$. Show that $R$ is an equivalence relation. Find the set of all lines related to the line $y = 2x + 4.$

asked Nov 8, 2012 by vaishali.a

1 answer

Show that the relation $R$ defined in the set $A$ of all polygons as $R = \{(P_1, P_2) :\: P_1\ and \;P_2$ have same number of sides$\}$, is an equivalence relation. What is the set of all elements in $A$ related to the right angle triangle $T$ with sides $3, 4$ and $5$?

asked Nov 8, 2012 by vaishali.a

1 answer

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 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?

asked Nov 8, 2012 by vaishali.a

1 answer

The relation $R$ in the set $A$ of points in a plane given by $R = \{(P, Q)$ : distance of the point $P$ from the origin is same as the distance of the point $Q$ from the origin $\}$, is

asked Nov 8, 2012 by vaishali.a

1 answer

Show that each of the relation $R$ in the set $A=\{x\in Z: 0 \leq x \leq 12\}$, given by $(i) R = \{(a,b) |a-b|$ is a multiple of 4 $\}$ is an equivalence relation. Find the set of all elements related to 1.

asked Nov 7, 2012 by vaishali.a

1 answer

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 $\{2, 4\}$.

asked Nov 7, 2012 by vaishali.a

1 answer

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.

asked Nov 7, 2012 by vaishali.a

1 answer

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.

asked Nov 7, 2012 by vaishali.a

1 answer

Check whether the relation $R$ in $R$ defined by $R = {(a, b) : a \leq b^3}$ is reflexive, symmetric or transitive.

asked Nov 7, 2012 by vaishali.a

1 answer

Show that the relation $R$ in $R$ defined as $R = {(a, b) : a \leq b} $, is reflexive and  transitive but not symmetric.

asked Nov 7, 2012 by vaishali.a

1 answer

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.

asked Nov 7, 2012 by vaishali.a

1 answer

Show that the relation $R$ in the set $R$ of real numbers, defined as $(R) =\{(a, b: (a \leq b^2 )\}$ is neither reflexive nor symmetric nor transitive.

asked Nov 7, 2012 by vaishali.a

1 answer

Let \[ F :R->R: f(x) =2x+1 \] and \[ g:R->R:g(x)=x^2 –2 \] find (I) gof (II) fog

asked Nov 2, 2012 by vaishali.a

0 answers

Show that function $ f\: N\ ->N$, given by $ f(x)\ =\ 2x$, is one – one.

asked Nov 2, 2012 by vaishali.a

0 answers

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Let $f:\{1,3,4\} \to \{1,2,5\} $ and $g:\{1,2,5\}\to\{1,3\}$ be given by $f = \{(1, 2), (3, 5), (4, 1)\}$ and $ g = \{(1, 3), (2, 3), (5, 1)\}$. Write down $gof$.

Let $f : R \to R$ be defined as $f (x) = 3x$. Choose the correct answer:

Let $f: R \to R $, be defined as $f(x) = x^4$. Choose the correct answer.

Let $A=R-\{3\}$ and $B=R-\{1\}$. Consider the function $ f:A \to B$ defined by $f(x)=\large\frac {x-2}{ x-3}$. Is $f$ one-one and onto?

Let \(A\) and \(B\)be sets. Show that\(f : A × B \to\; B×A\)such that \( f(a,b)\,=\,(b,a)\) is bijective function

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.(i) \(f : R\; \to R\; defined \; by \; f(x)\; =\; 3-4x\)

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.

Show that the Signum Function \(f : R \to R\), given by \[ f(x) = \left\{ \begin{array}{l l} 1, & if \; x \; > 0 \\ 0, & if\; x\;= 0 \text{ is neither one-one nor onto } \\ -1, & if\; x \;< 0 \end{array} \right. \]

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.

Prove that the Greatest Integer Function \(f : R \to R\), given by \(f (x) = [x]\), is neither one-one nor onto, where \([x]\) denotes the greatest integer less than or equal to \(x\).

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

Show that the function \(f : R_* \rightarrow R_* \) defined by \( f(x) = (\frac{1} {x})\) is one-one and onto, where \(R_* \) is the set of all non-zero real numbers. Is the result true, if the domain \(R_* \) is replaced by \(N\) with co-domain being same as \(R_*\)?

Let $R$ be the relation in the set $N$ given by $R=\{(a,b): a=b-2, b>6\}$. Choose the correct answer:

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.

Let \(L\) be the set of all lines in \(XY\) plane and \(R\) be the relation in \(L\) defined as \(R = \{(L1, L2) : L1\:\) is parallel to \(L2\}\). Show that \(R\) is an equivalence relation. Find the set of all lines related to the line \(y = 2x + 4.\)

Show that the relation \(R\) defined in the set \(A\) of all polygons as \(R = \{(P_1, P_2) :\: P_1\ and \;P_2\) have same number of sides\(\}\), is an equivalence relation. What is the set of all elements in \(A\) related to the right angle triangle \(T\) with sides \(3, 4\) and \(5\)?

The relation $R$ in the set $A$ of points in a plane given by $R = \{(P, Q)$ : distance of the point $P$ from the origin is same as the distance of the point $Q$ from the origin $\}$, is

Show that each of the relation $R$ in the set $A=\{x\in Z: 0 \leq x \leq 12\}$, given by $(i) R = \{(a,b) |a-b|$ is a multiple of 4 $\}$ is an equivalence relation. Find the set of all elements related to 1.

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.

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.

Check whether the relation $R$ in $R$ defined by $R = {(a, b) : a \leq b^3}$ is reflexive, symmetric or transitive.

Show that the relation \(R\) in \(R\) defined as \(R = {(a, b) : a \leq b} \), is reflexive and  transitive but not symmetric.

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.

Show that the relation \(R\) in the set \(R\) of real numbers, defined as $(R) =\{(a, b: (a \leq b^2 )\}$ is neither reflexive nor symmetric nor transitive.

Let \[ F :R->R: f(x) =2x+1 \] and \[ g:R->R:g(x)=x^2 –2 \] find (I) gof (II) fog

Show that function \( f\: N\ ->N\), given by \( f(x)\ =\ 2x\), is one – one.