Which of the following relations on the set $A = \{1, 2, 3\}$ are equivalence ?

(A) $R = \{(1,1), (2,2), (1,2), (2,1)\}$

(B) $R = \{(1,1), (2,2), (3,3)\}$

(C) $R = \{(1,1), (1,2), (2,1)\}$

(D) $R = \{(1,1), (2,2), (3,3), (1,2), (2,1), (2,3), (3,2), (1,3), (3,1)\}$

(E) $R = \{(1,1), (2,2), (3,3), (1,2)\}$

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Let N be set of natural numbers, then the function $f: \mathbb{N} \rightarrow \mathbb{N}$, defined by

$f(x) = \begin{cases} \frac{n+1}{2} & \text{, if } n \text{ is odd} \\ \frac{n}{2} & \text{, if } n \text{ is even} \end{cases}$, is

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

Let $A = \{a, b, c\}$. Then number of relations containing $(a,b)$ and $(b, c)$ which are reflexive and transitive but not symmetric is

Which of the following statements are true?

(A) $f:\mathbb{R} \to \mathbb{R}$ given by $f(x) = 3x$ is one-one onto.

(B) $f:\mathbb{R} \to \mathbb{R}$ given by $f(x) = x^4$ is one-one and onto.

(C) $f:\mathbb{Z} \to \mathbb{Z}$ given by $f(x) = x^2$ is neither one-one nor onto.

(D) $f:\mathbb{R} \to \mathbb{R}$ given by $f(x) = |x|$ is neither one-one nor onto.

(Where $\mathbb{R}$ is the set of all real numbers and $\mathbb{Z}$ is the set of all integers)

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A function $f: \mathbb{R} \rightarrow \{x \in \mathbb{R}: -1 < x < 1\}$ is defined as $f(x) = \frac{x}{1+|x|}$, then $f$ is: