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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?

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    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
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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.

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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.

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    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

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