What is the smallest closed subset of Z containing 2 and 0?

[Gear300]

The question given is: Determine the smallest subset A of Z such that 2 ε A and A is closed with respect to addition.

The answer given was the set of all positive even integers, but I was thinking that the smallest subset would be the given element and the identity element (0 in this case) so that A = {0,2}...wouldn't this be more accurate?

 

[Tac-Tics]

To understand why that is not correct, you have to know what closure of an operator over a set means.

A set S is closed under a binary operator + iff for all x, y in S, x + y is in S.

A = {0, 2} isn't closed under addition because the definition is not satisfied. We can find a counter example where x and y are both in S, but x + y is not. The counter example is with x = 2 and y = 2.

 

[AUMathTutor]

Closed means you can do it to any two (possibly non-unique) elements and get an answer in your set.

S = {0, 2} doesn't work since 2+2=4 is not in S.

 

[Gear300]

I see. Thanks...funny I didn't notice that.