Question

Closure illustrates that a characteristic of a set is not

necessarily a characteristic of all of its subsets. The real
numbers are closed with respect to multiplication, but the irrational numbers, a subset of the real numbers, are not. Give an example of a set that is not mathematical that has a particular characteristic, but which has a subset without this characteristic.

Answer 1

There are many examples to pick from, but one example is this:

The set of rational numbers (aka any fraction of two integers) is closed under the operation of division. Divide any two rational numbers and we get some other rational number.

However, the set of integers is not closed under division. If we divided 10 over 3, then we get 10/3 = 3.333 approximately which isn't an integer. So just because the set of integers is a subset of the rationals, it doesn't mean that the idea of closure follows suit from superset to subset.

Side note: The term "superset" is basically the reverse of a subset. If A is a subset of B, then B is a superset of A.