Proofs about a group property

Question

I have a set $B$,and for $b$ belonging to $B$,there exists a reverse $b$. Is it enough if I prove this to be true for $1$ element of the set, then it must be true for the whole set of $B$, or must I prove this property $\forall x$ that belong to $B$.

Answer 1

If you want to show that $B$ is a group, you need to show that every element of $B$ has an inverse. You need to be able to calculate with any group element, so every element must have the basic properties given in the group axioms.