Given the relation $R=\{(1,2),(2,3)\}$ on a set $A=\{1,2,3\}$, then which of following statements are TRUE?

A. Minimum number of ordered pairs are added to R so that enlarged relation is reflexive is 3

B. Minimum number of ordered pairs are added to R so that enlarged relation is symmetric is 2

C. Minimum number of ordered pairs are added to R so that enlarged relation is an equivalence relation is 7

D. Minimum number of ordered pair are added to R so that enlarged relation is an equivalence relation is 6.

Choose the correct answer from the options given below:

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

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

A. $f$ is one-one

B. $g$ is one-one

C. $f + g$ is one-one

D. $f + g$ is not one-one

Choose the correct answer from the options given below:

A. A relation $R$ on a set $A$ is called an equivalence relation, if it is reflexive, symmetric and transitive.

B. The function $f : R \to R$ defined by $f(x) = e^x$ is not one-one.

C. The one-one function is also known as injective function.

D. The onto function is also known as subjective function.

E. A function $f : X \to Y$ is said to be many-one, if two or more than two elements in set $X$ have the different image in set $Y$.

Choose the correct answer from the option given below:

The function $f: \mathbb{R} \rightarrow [-1, 1]$ defined by $f(x) = \cos x$ is:

Let A= {1, 2, 3}, then the possible equivalence relations on A are:

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

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

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

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

Choose the correct answer from the options given below:

The relation R in $\mathbb{R}$ (set of real numbers) is defined by $R = \{(a,b): a \leq b^3\}$, then R is