Question

is the set {-3,0,3} closed under addition, subtraction, multiplication, or devision?

Answer 1

Call your set \(\mathbb{S}=\{ -3, 0, 3\}\)
Addition:
\(-3+0=-3 \in \mathbb{S}\)
but
\(3+3=6\not\in \mathbb{S}\)
so not closed under addition.

Similarly:
\(-3-3=-6\not \in \mathbb{S}\)
not closed under subtraction

Multiplication:
\(-3 \times 3 = -9 \not\in \mathbb{S} \)

Division:
\(-3 \div 3 = -1 \not \in \mathbb{S}\)

For the set to be closed under those operations, you must guarantee that every element in that set, when an operation is performed on an another element in the set, is still an element in this set. This must work for ALL elements in the set. The above shows counterexamples for each operation, showing it is not closed under any operation.