I am asked to prove that when squaring any element of the empty set, one should always get zero.
Of course the empty set is the set which contains no elements. If you square nothing then you should get nothing, or equivalently zero. I am having an issue as to how to show this formally.
I'm not even sure on which form of proof I should try to show this, i.e., direct proof, contradiction, or maybe contraposition.
Maybe for contradiction I could say something along the following:
Let $x \in A :x \; \ne0.$ Then $x^2 \ne 0.$ Hence, $A \ne \emptyset.$ We then have arrived at a contradiction!
Does this even make sense?
Maybe I could say something along the lines like: Let $ x \in A \cap B$, where A and B are two disjoint sets. Then try to prove in a direct fashion.
Any guidance or ideas would be much appreciated. Thank you