Assertion (A): Let $f(x) = e^{x}$ and $g(x) = \log x$. Then $(f + g)x = e^{x} + \log x$ where domain of $(f + g)$ is $\mathbb{R}$.

Let $f : \mathbb{R} - \left\{ \frac{4}{3} \right\} \to \mathbb{R}$ be a function defined as:$$f(x) = \frac{4x}{3x+4}$$Show that $f$ is a one-one function. Also, check whether $f$ is an onto function or not.

If $f:R^{+}\rightarrow R$ is defined as $f(x) = \log_{a} x$ ($a > 0$ and $a\ne1$), prove that f is a bijection. ($R^{+}$ is a set of all positive real numbers.)

A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever. Let $I$ be the set of all citizens of India who were eligible to exercise their voting right in the general election held in 2019. A relation $R$ is defined on $I$ as follows: $R = \{(V_1, V_2): V_1, V_2 \in I \text{ and both use their voting right in general election 2019}\}$. 1. Two neighbors $X$ and $Y \in I$. $X$ exercised his voting right while $Y$ did not cast her vote in general election 2019. Which of the following is true? (a) $(X,Y) \in R$ (b) $(Y,X) \in R$ (c) $(X,X) \notin R$ (d) $(X,Y) \notin R$ 2. Mr. '$X$' and his wife '$W$' both exercised their voting right in general election 2019. Which of the following is true? (a) both $(X,W)$ and $(W,X) \in R$ (b) $(X,W) \in R$ but $(W,X) \notin R$ (c) both $(X,W)$ and $(W,X) \notin R$ (d) $(W,X) \in R$ but $(X,W) \notin R$ 3. Three friends $F_1$, $F_2$, and $F_3$ exercised their voting right in general election 2019, then which of the following is true? (a) $(F_1,F_2) \in R$, $(F_2,F_3) \in R$ and $(F_1,F_3) \in R$ (b) $(F_1,F_2) \in R$, $(F_2,F_3) \in R$ and $(F_1,F_3) \notin R$ (c) $(F_1,F_2) \in R$, $(F_2,F_2) \in R$ but $(F_3,F_3) \notin R$ (d) $(F_1,F_2) \notin R$, $(F_2,F_3) \notin R$ and $(F_1,F_3) \notin R$ 4. The above defined relation $R$ is: (a) Symmetric and transitive but not reflexive (b) Universal relation (c) Equivalence relation (d) Reflexive but not symmetric and transitive 5. Mr. Shyam exercised his voting right in General Election 2019, then Mr. Shyam is related to which of the following? (a) All those eligible voters who cast their votes (b) Family members of Mr. Shyam (c) All citizens of India (d) Eligible voters of India

A relation R is defined on $N\times N$ (where N is the set of natural numbers) as: $(a, b)~R~(c,d)\Leftrightarrow a-c=b-d$ Show that R is an equivalence relation.