What does it mean for a set to be contained in another?

Question

If $A$ & $B$ are two sets, what does '$A$ is contained in $B$' mean ? Does it mean that $A$ is a subset of $B$ or $A$ is a proper subset of $B$?

Answer 1

It's a bit ambiguous, but hopefully it's only used when there is little chance of confusion. If there is a chance, it is better to use notation. Either $A\in B$ for $A$ being an element of $B$, or $A\subseteq B$ for $A$ being a subset of $B$.

If it's the latter, generally it is not necessarily a proper subset. Authors however can do whatever they want, as long as they define it. You have to watch out for nonstandard language and notation.

Answer 2

The only thing that "$A$ is contained in $B$" means is that the implication

$$x \in A \implies x \in B $$

is satisfied. It could be a proper or non-proper subset. Often, when the subset is proper, most authors explicitely state this.