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A relation $R$ on set of real numbers $(\mathbb{R})$ is defined as $R = \{(a,b): a \le b, a,b \in \mathbb{R}\}$. The relation is :

The function $f: R \to [0,1]$ defined by $f(x) = \dfrac{x^2}{x^2+1}$ is: (Where $R$ is set of real numbers)

A. One-one

B. one-one and onto

C. onto

D. Not one-one

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The greatest integer function $f: R\to R$ defined by $f(x) = [x]$, then which of the following are true?

(A) $f(x)$ is one - one and onto.

(B) Domain of $f(x) = R$ (the set of real numbers).

(C) $f(x)$ is neither one-one nor onto.

(D) Range of $f(x) = Z$ (The set of integers).

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If n(A) = 3 and n(B) = 2, then number of non-empty relations from A to B is

The relation R in the set of real numbers, defined as R = {(a, b): 1 + ab > 0}, is:

The signum function $f:R\to R$, defined by

$f(x)=\begin{cases}-1, & x<0\\0, & x=0\\1, & x>0\end{cases}$ is: (Where R is set of real numbers)

For the function $f(x) = |x-1| + |x-2| + |x-3|$

Match the LIST-I with LIST-II

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Let S be the set of all real numbers and let R be a relation on S, defined by $aRb \Leftrightarrow |a-b| \leq 1$. Then, R is

Match the LIST-I with LIST-II (Consider N and R as set of natural numbers and set of real numbers respectively)

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Let $R_1, R_2, R_3, R_4$ be the relations defined on the set $A=\{1,2,3\}$. then

Match List-I with List-II

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Consider a function $f:[-1,1] \to Y$ defined as $f(x)=\dfrac{x}{x+2}$ and $Y=range(f(x))$, then which of following statements are correct?

(A) Function $f(x)$ is one-one

(B) Function $f(x)$ is not one-one

(C) Function $f(x)$ is onto

(D) Function $f(x)$ is invertible

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Let $A = \{ 1,2,3 \}$, then Match List-I with List-II

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If $f\left(x+\dfrac{1}{x}\right) = x^2 + \dfrac{1}{x^2}$ then $f(x) =$

Let $f: \mathbb{N} \to \mathbb{N}$ defined by $f(x) = 3x^2 + 2x + 7$, where $\mathbb{N}$ is set of natural numbers. Then $f(x)$ is:

For the relations $R_1, R_2, R_3, R_4$ defined on the set $A = \{1,2,3\}$,

Match the LIST-I with LIST-II

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Consider a function $f:\left[0,\frac{\pi}{2}\right] \to R$, given by $f(x)=\sin x$ and $g:\left[0,\frac{\pi}{2}\right] \to R$, given by $g(x)=\cos x$, Where R is a set of real numbers, Which of the following statements are correct?

(A) f and g are one-one

(B) f and g are not one-one

(C) f+g is not one-one

(D) f+g is one-one

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The relation R on the set A = {1, 2, 3} defined by R = {(1,1), (2,2), (3,3), (1,3), (2,3), (1,2)} is

Let X be a non empty set and S be the collection of all subsets of X. A relation R in S is defined by $R=\{(A,B): A\subseteq B\}$, where $X\subseteq Y$ means $X$ is proper subset of $Y$. Then R is:

Let a function $f:\mathbb{R}\to\mathbb{R}$ defined as $f(x)=x-[x]$, (where $\mathbb{R}$ is set of real numbers & $[.]$ denotes greatest integer function) .Then the function:

If a function $f: R \to T$, defined by $f(x) = \sqrt{3}\sin x + \cos x$ is an onto function, then the interval of $T$ is

The relation R on the set N of natural numbers defined as $R = \{(x,y): y = x+3 \text{ and } x \le 5\}$ is

The number of symmetric relations on set $A = \{1,2,3\}$ is

If $A = R - \{2\}$ and $B = R - \{1\}$ and function $f: A \rightarrow B$ defined by $f(x) = \frac{x-1}{x-2}$, then $f$ is

Match the LIST-I with LIST-II

[Where R and Z denotes set of real numbers and set of integers respectively]

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Let X be a non-empty set and let P be the collection of all subsets of X. Let R be a relation in P, defined by $R = \{(A, B): A \text{ is subset of } B\}$. Then, R is

A. Reflexive relation

B. Symmetric relation

C. Not a transitive relation

D. Transitive relation

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For the functions $f(x) = x^2$ and $g(x) = x^3$, which of the following statements are TRUE?

[where N is the set of natural numbers and Z is the set of integers]

(A) $f: N \to N$ is one-one but not onto.

(B) $f: Z \to Z$ is neither one-one nor onto.

(C) $g: N \to N$ is one-one but not onto.

(D) $g: Z \to Z$ is one-one but not onto.

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Let $A = \{0, 1, 2, 3\}$ and R be a relation on set A defined as $R = \{(0,0), (0,1), (0,3), (1,0), (1,1), (2,2), (3,0), (3,3)\}$, then the relation R is

For the relation R defined below, where $\mathbb{R}$ and N denotes set of real numbers and set of natural numbers respectively, Match the LIST-I with LIST-II

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If $f(x) = \sin^{-1}\left(\dfrac{2^{x+1}}{1+4^x}\right)$, then

Match List-I with List-II

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Let a relation $R$ be defined on a set $A = \{1,2,3,4,5\}$ as $R = \{(a,b): |a-b| \text{ is even or zero}\}$. Then which of the following statements is not TRUE?