Example 1: Closure and the Set of Integers
Take the set of all
integers
{..., 2, 1, 0, 1, 2, ...}
and the operation of addition (+). If we add any two integers, do we always get an integer?
If we add 5 and 3, we get 8, which is an integer. The answer
is yes, so we can say that the set of integers is closed with respect to addition.
Now consider the set of integers and multiplication. Is the set of integers closed
with respect to multiplication? Can you think of any two integers that, when multiplied,
give something that is not an integer? There is none, so the set of integers is closed
with respect to multiplication.
Now consider the set of integers and division (÷). Are there
any two integers that, when divided, do
not give an integer? Try 1 and 2.
1÷2 = 0.5. 0.5 is not
an integer, so we say that the set of integers is not closed with respect to division.
