Answer: False
Reason
Pick any two negative numbers you want. Divide them. Do you get a negative result? No, you shouldn't. The two negatives cancel to get a positive result.
Example: -10 ÷ -5 = 2
We only need one counterexample to disprove a claim.
The fact we do not get a negative result means the set of negative numbers is not closed under division.
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Here's an example of something that is closed.
The set of positive integers is closed under addition. Why? Pick any two positive whole numbers. Add them. The result is some other positive number. We cannot leave the set when applying addition to two random items.
Examples:
It is impossible to pick two whole numbers, add them, and somehow jump outside the set of positive integers.