# How to make this hypothesis in a negative form?

I'm learning conditional statements in Math and I am tasked to make the hypothesis and the conclusion rewritten in a negative form.

This one is where I have trouble with:

If a polygon has exactly four sides, then it is a quadrilateral.

I'm trying to find a place where to put the "not" or "no" in the hypothesis. I already figure out what the conclusion should look like:

... then it is not a quadrilateral.

How do I make the hypothesis negative?

• If a polygon doesn't have exactly four sides, then it isn't a quadrilateral. – Andrew Tobilko Apr 8 at 8:44
• @AndrewTobilko That makes sense. Can that be an answer? – Water Water Apr 8 at 8:44
• @AndrewTobilko Should I answer that instead? – Water Water Apr 8 at 8:49
• yes, feel free because I still don't quite get what your struggles were – Andrew Tobilko Apr 8 at 8:50

The existing answer is correct, but a second correct answer is to use "unless":

• Unless a polygon has exactly four sides, it is not a quadrilateral.

We could also say:

• A polygon is a quadrilateral only if it has exactly four sides.

If we wanted to state in a single sentence both that four-sided polygons are quadrilaterals and that non-four-sided ones aren't, we could say:

• A polygon is a quadrilateral if and only if it has exactly four sides.

This option is a little cumbersome but is often seen in formal definitions. "Only if" on its own (without "if and") would probably be interpreted the same way, but may be open to interpretation.

The simplest way to make the hypothesis and conclusion negated would be:

If a polygon does not have exactly four sides, then it is not a quadrilateral.

For a shortened version:

If a polygon doesn't have exactly four sides, then it isn't a quadrilateral.

• I think given this is a formal statement you would want to avoid contractions. "If a polygon does not have exactly four sides, then it is not a quadrilateral." – Daniel Roseman Apr 8 at 9:10