Problem: Let the set consisting of the squares of the positive integers be called ; thus is the set . If a certain operation on one more members of the set always yields a member of the set, we say that the set is closed under that operation. Then is closed under:
Answer Choices:
A.
B.
C.
D. extraction of a positive integral root
E. none of these
Solution:
is not closed under addition since, for example, and , not being a perfect square of an integer, is not a member of . Similarly for division and positive integral root extraction. is closed under multiplication because, if and are elements of , is a member of ( contains the squares of all positive integers and is an integer.