# A “positive statement” in mathematical proof

definitionproof-writingterminology

I'm going through How to Prove It: A Structured Approach by Daniel J. Velleman and some terms that I frequently see are "positive statement" and "negative statement". I'm not sure what these are referring to and when I research the terms I can't seem to find definitions that relate to mathematical proof.

If you were tasked with defining these terms, how would you define them?

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#### Best Answer

In this situation, it seems that a negative statement is a proposition that comes with a logic negation in front of it. Let me try to explain it with an example, that should be clearer:

Positive statement: “There exists a function $$f:X \rightarrow Y$$ such that $$f(x)$$ is constant on $$X$$

Negative statement: “There does not exist a function $$f:X \rightarrow Y$$ such that $$f(x)$$ is constant on $$X$$“, that logically means “for every function $$f:X \rightarrow Y$$, $$f(x)$$ is not constant on $$X$$ (that is there exists at least an $$a$$ and $$b$$ both in $$X$$ such that $$f(a) \neq f(b)$$.