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Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then R is

(a) Equivalence

(b) Reflexive and symmetric

(c) Transitive and symmetric

(d) Reflexive, transitive but not symmetric

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Let P ={1,2,4,5}. The total number of distincts relations that can be defined on P is

(a) 16

(b) $2^{16}$

(c) 32

(d) 256

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The function $f:[0,\infty)$ -> R given by $f(x)= \frac {x}{x+1}$ is

(a) one -one and onto

(b) one-one but not onto

(c) onto but not one-one

(d) Neither one-one nor onto

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Let f : R -> R be defined by $f(x) = 2x - 1. Then $f^{–1}(x)$ is given by

(a) $\frac {x-2}{2}$

(b) $\frac {x+1}{2}$

(c) $\frac {x-1}{2}$

(d) $\frac {2x-2}{3}$

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The function f:R-> R given by $f(x)= \frac {x^2 -8}{x^2+2}$ is

(a) one -one and onto

(b) one-one but not onto

(c) onto but not one-one

(d) Neither one-one nor onto

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let A={1,2,3} Which of the following relation defined on A is a reflexive relation?

(a) {(1, 1), (1, 2), (2, 1), (2, 2)}

(b) {(1, 2), (2, 1)}

(c) {(1, 1), (2, 2), (3, 3)}

(d) {(1, 2), (2, 3), (3, 1)}

Answers

Which of the following is an onto function f:R -> R?

(a) $f(x) = x^2$

(b) $f(x) = 2x + 1$

(c) $f(x) = |x|$

(d) $f(x) = x^2 + 1$

Answers

For real numbers x and y, define xRy if and only if $x – y + \sqrt 2$ is an irrational number. Then the relation R is

(a) symmetric

(b) Reflexive

(c) Transitive

(d) None of these

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$f(x) =x^2 +1$ and $g(x) = sin x$, then the value of $g \circ f$ is

(a) $sin^2x + 1$

(b) $sin (x^2 +1)$

(c) $x sin x + 1$

(d) None of these

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let f(x)=[x] and g(x) =|x|, the the value of $(g \circ f) (\frac {-5}{3}) - (f \circ g)(\frac {-5}{3}) $ is

(a) 1/2

(b) 1

(c) -1

(d) -2

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(i) Let the relation R be defined in N by aRb if 2a + 3b = 30. Then R = _____

(ii) if $f(x)=x^2 +2$ and $g(x)= 1 - \frac{1}{1-x}$, the $f \circ g$ is _____

(iii) The inverse of the function $f(x)= \frac {x}{x+2}$ is _______

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Let f : W -> W be defined as

$f(n)=\begin{cases} & \text{ n-1 if } n \ is \ odd \\ & \text{ n+1 if } n \ is \ even \end{cases}$

Then show that f is invertible.Also, find the inverse of f.

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Show that f : [– 1, 1] -> R , given by $f(x) =\frac {x}{x+2}$ is one-one.

Find the inverse of the function f : [– 1, 1] -> Range of f

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Show that the relation R in the set N X N defined by (a,b)R(c,d) if $a^2+d^2 =b^2+ c^2$ for all a,b,c,d in N is an equivalence relation

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Functions f , g : R -> R are defined, respectively, by $f(x) = x^2 + 3x + 1$,g(x) = 2x – 3, find

(i) $f \circ g$

(ii) $g \circ f$

(iii)$f \circ f$

(iv) $g \circ g$

Answers

Show that the relation R in the set A X A defined by (a,b)R(c,d) if $a+d =b+c$ for all a,b,c,d in A is an equivalence relation. Here A={1,2,3...10}

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If $f(x) =\frac {4x+3}{6x-4}$ and $x \ne \frac {-2}{3}$ , show that fof(x) = x for all x $x \ne \frac {-2}{3}$ . Also, find the inverse of f

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Let A= R -{-1} and * be the binary operation on A defined by

a*b = a+b+ab

(i) Show that * is commutative and Associative

(ii) Find the identity element for * on A

(iii) Prove that every element of A is invertible

Answers

Prove that the function f:[0, $\infty$) -> R given by $f(x) = 9x^2+ 6x – 5$ is not invertible. Modify the codomain of the function f to make it invertible, and hence find $f^{–1}$

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Check whether the relation R in the set R of real numbers, defined by R = {(a, b) : 1 + ab > 0}, is reflexive, symmetric or transitive

Show that the relation R on the set Z of integers, given by R = {(a, b) : 2 divides (a – b)} is an equivalence relation

Prove that the relation R in the set A = {1, 2, 3, 4, 5, 6, 7} given by R = {(a, b) : |a – b| is even} is an equivalence relation

Show by examples that the relation R in , defined by

R = {(a, b) : $a \le b^3$} is neither reflexive nor transitive.

Let R be a relation on the set A of ordered pairs of positive integers defined by (x, y) R (u, v) if and only if xv = yu. Show that R is an equivalence relation

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