How can I show that this function is injective, surjective or bijective?

Question

I have this function $f(n) = \lceil \frac{n}{2} \rceil $ and I want to proof if it is injective surjective or bijective. Maybe I don't know how to do it because I don't know how to define this function.

Answer 1

$f(1) = \lceil \frac{1}{2} \rceil = 1$

$f(2) = \lceil \frac{2}{2} \rceil = 1$

So $f(1)=f(2)$, and therefore $f$ is not injective.

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Let $n\in\mathbb{Z}$.

Note that $f(2n) = \lceil \frac{2n}{2} \rceil = n$, then $f$ is surjective.

(I suppose that the domain and codomain of $f$ is $\mathbb{Z}$)

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How $f$ is not injective, then $f$ is not bijective.

Answer 2

$\lceil x \rceil$ is the smallest integer greater than or equal to $x$.

It's probably most helpful to draw a picture (taken from Wikipedia):

Now, you should ask yourself:

• Can you find distinct numbers $a$ and $b$ such that $\lceil a \rceil = \lceil b \rceil$? If so, the function is not injective.
• Can you find a number $y$ such that for all $x$, $\lceil x \rceil \neq y$? If so, the function is not surjective (assuming the codomain is $\mathbb{R}$).