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I recently had a discrete math midterm that I am trying to review, and one of the topics that really has me confused is the idea of proving uniqueness.

Let $x \in \mathbb R$. Prove that the $\lceil{x}\rceil$ is unique; that is, prove that there is at most one $n \in \mathbb Z$ with $n − 1 < x ≤ n.$

My professor, when doing these problems in the past, tended to use $x'$ as a second variable when proving uniqueness, but I am struggling with comprehending the logic behind these proofs. I understand the structure, i.e. that you are trying to prove that $x, x'$ both make the predicate true and that implies that $x=x'$, but I can't execute any of these proofs.

Also, please note that I am not trying to prove the existence here, but instead uniqueness.

Any help would be greatly appreciated.

M. Silber
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  • I saw that post, but that seemed as though it was trying to prove existence, not uniqueness, albeit on a similar problem. I could be reading the post incorrectly, but I posted this one because it seemed like a different problem to me. – M. Silber Mar 15 '18 at 13:39
  • Yeah, I can say after reviewing the linked post extensively that the questions are asking for different things. Where as proving existence involves proving that there is at least one x that makes the condition true, whereas uniqueness proves that there is at most one x that makes the condition true, and they have distinct proof structures. – M. Silber Mar 15 '18 at 14:39
  • You can see proof-that-floor-function-is-well-defined. – Mauro ALLEGRANZA Mar 15 '18 at 15:20

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