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Is the following function surjective from the set of integers to the set of integers? $$\lceil x/2\rceil$$ My initial intuition says that it is, but I don't know if once the element $x$ from the domain starts getting higher in value (when $x$ approaches infinity), it would eventually miss an integer.

I hope this makes sense.

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  • $\begingroup$ What happens if $x=2n,$ $n$ an integer? $\endgroup$ Commented Jul 23, 2018 at 15:13
  • $\begingroup$ @ChrisLeary you get the double value of n. $\endgroup$
    – KetDog
    Commented Jul 23, 2018 at 15:17
  • $\begingroup$ I believe you should get $n.$ Check Parclay's answer below. $\endgroup$ Commented Jul 23, 2018 at 16:35
  • $\begingroup$ @ChrisLeary yes, I understand now. If 2*n = x in the original equation. $\endgroup$
    – KetDog
    Commented Jul 23, 2018 at 16:45

2 Answers 2

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Notice that $\lceil (2n)/2\rceil = n$.

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For any integer $y$ pick $x=2y$, which is an integer. Then $x/2$ is an integer, and since the ceiling of an integer is itself, $\lceil x/2\rceil=y$. Therefore the function has a preimage for every element in its codomain and is surjective.

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