Sets, Relations and Functions
If $$f:A\rightarrow B $$ is surjective then
Show that the Signum function $$f:R \rightarrow R$$, given by $$\displaystyle f(x)=\begin{cases}1,\ if\ x > 0 \\0,\ if\ x = 0 \\ -1,\ if\ x < 0 \end{cases}$$. is neither one-one nor onto.
$$\displaystyle f(x)= \left\{\begin{matrix}1&\text{if} x > 0 \\0 &\text{if} x = 0 \\ -1& \text{if} x < 0 \end{matrix}\right\}$$.
It is seen that $$f(1)=f(2)=1$$, but $$1 \neq 2$$.
$$\therefore f$$ is not one-one.
Now, as $$f(x)$$ takes only $$3$$ values $$(1,0,$$ or$$-1)$$ for the element $$-2$$ in co-domain $$R$$,
there does not exist any $$x$$ in domain $$R$$ such that $$f(x)=-2$$
$$\therefore f$$ is not onto
Hence, the signum function is neither one-one nor onto.
If $$f:A\rightarrow B $$ is surjective then
Which of the following is an onto function
If $$A =\{1, 2, 3\}$$ and $$ B = \{4, 5\}$$ then the number of function $$f : A \rightarrow B$$ which is not onto is ______
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Show that the function $$f: R\rightarrow R:f(x)=\sin x$$ is neither one-one nor onto.
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If the function $$f : R \rightarrow$$ A given by $$f(x)\, =\, \displaystyle \frac{x^{2}}{x^{2}\, +\, 1}$$ is a surjection, then A is
Give one example of each of the following function : Onto but not one-one
Let $$E=\{1, 2, 3, 4\}$$ and $$F=\{1, 2\}$$ then the number of onto functions from E to F is