Consider the following statements:
$(i)$ $A$ relation is a special case of a function.
$(ii)$ $A$ function is a special case of a relation.
$(iii)$ Both relation and function are the same.

Let $f(x) = x^{2}$ and $g(x) = 2x + 1$ be two real functions. Find $(f+g)(x)$,$(f-g)(x)$,$(fg)(x)$,and $(\frac{f}{g})(x)$.

Let $f: R \to R$ be a function. Define $g: R \to R$ by $g(x) = |f(x)|$ for all $x$. Then $g$ is

If $f(a) = \log \left| \frac{1-a}{1+a} \right|$ for $a \neq \{-1, 1\}$,then the set of values of all $a$,for which $f\left( \frac{2a}{1+a^2} \right) > 0$ is

The number of non-constant functions $f: X \to Y$ where $X = \{0, 1, 2\}$ and $Y = \{1, 2, 3, 4, 5, 6, 7, 8\}$ such that $f(i) \leq f(j)$ whenever $i < j$ is:

Answer the following by appropriately matching the lists based on the information given in the paragraph.
Let $f(x) = \sin(\pi \cos x)$ and $g(x) = \cos(2\pi \sin x)$ be two functions defined for $x > 0$. Define the following sets whose elements are written in increasing order:
$X = \{x : f(x) = 0\}, Y = \{x : f'(x) = 0\}$
$Z = \{x : g(x) = 0\}, W = \{x : g'(x) = 0\}$
$List-I$ contains the sets $X, Y, Z$ and $W$. $List-II$ contains some information regarding these sets.

$List-I$	$List-II$
$(I) X$	$(P) \supseteq \{\frac{\pi}{2}, \frac{3\pi}{2}, 4\pi, 7\pi\}$
$(II) Y$	$(Q) \text{ an arithmetic progression}$
$(III) Z$	$(R) \text{ NOT an arithmetic progression}$
$(IV) W$	$(S) \supseteq \{\frac{\pi}{6}, \frac{7\pi}{6}, \frac{13\pi}{6}\}$
	$(T) \supseteq \{\frac{\pi}{3}, \frac{2\pi}{3}, \pi\}$
	$(U) \supseteq \{\frac{\pi}{6}, \frac{3\pi}{4}\}$

$(1)$ Which of the following is the only $CORRECT$ combination?
$(1) (II), (R), (S)$ $(2) (I), (P), (R)$ $(3) (II), (Q), (T)$ $(4) (I), (Q), (U)$
$(2)$ Which of the following is the only $CORRECT$ combination?
$(1) (IV), (Q), (T)$ $(2) (IV), (P), (R), (S)$ $(3) (III), (R), (U)$ $(4) (III), (P), (Q), (U)$