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A set of numbers is said to be closed under an operation if the result of combining any two numbers in the set results in a number that is also in the set. Decide whether or not each set is closed under the operation.

a. $\{\text{positive integers}\}$; division

b. $\{\text{odd integers}\}$; multiplication

c. $\{\text{odd integers}\}$; addition

d. $\{\text{integers ending in $4$ or $6$}\}$; multiplication

Thank you in advance.

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    $\begingroup$ Where are you facing difficulty? Your definition seems straightforward. Try to play with the set and the given operation to reach an answer. $\endgroup$
    – Shobhit
    Commented Nov 9, 2019 at 9:46
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    $\begingroup$ To show a set is $not$ closed under an operation, it suffices to exhibit a single exception. For a., $1$ & $2$ are positive integers but $1/2$ isn't. $\endgroup$
    – DanielWainfleet
    Commented Nov 9, 2019 at 9:56

1 Answer 1

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Hint: try to find examples for exceptions to each of these rules. So, if you can find a single example in a set of numbers that corresponds to a value outside of the set, then the set is not closed. As an example, let's look at part a.

a. {positive integers}; division

Numbers that are on some "edge" of the set are a good place to start. The smallest number in the positive integers set is 1, so start there. Divide 1 by 1 and you get an integer, but divide 1 by some positive integer greater than 1, then you get a number less than 1, which is not a positive integer, so the first set is not closed under division. Use a similar approach for the other parts of the question.

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