R D Sharma Solutions for Chapter: Function, Exercise 1: EXERCISE

If $f :A\to B$ is an injection, such that range of $f=\left\{a\right\}$, determine the number of elements in $A$.

Let $A=[-1, 1].$ Then, discuss whether the function from $A$ to itself is one-one, onto or bijective : $f \left(x\right)=\frac{x}{2}$

Let $A=\left\{1, 2, 3\right\}$. Write all one-one from $A$ to itself.

Show that the exponential function$f:R\to R$, given by $f\left(x\right)=e^x$, is one-one but not onto. What happens if the co-domain is replaced by ${R_0}^{+}$(set of all positive real numbers)?

Give examples of two surjective functions $f_1$ and $f_2$ from $Z$ to $Z$ such that $f_1+f_2$ is not surjective.

Show that if $f_1$ and $f_2$ are one-one maps from $R$ to $R,$ then the product $f_1\times f_2:R\to R$ defined by $(f_1\times f_2)\left(x\right)=f_1\left(x\right)f_2\left(x\right)$ need not be one - one.

Given $A=\left\{2,3,4\right\}, B=\left\{2,5,6,7\right\}$. Construct an example of a mapping from $A$ to $B$ which is not injective

Given $A=\left\{2,3,4\right\} B=\left\{2,5,6,7\right\}$ Construct an example of a mapping from $B$ to $A$.