About definition of superset

Question

I have read definition of superset somewhere as "a set containing all elements of a smaller set.". This implies that for set A to be a superset of B, B must be a proper subset of A, that is, A must contain at least one element which is not in B. Does it mean that a set A cannot be a superset of itself? or does there exist terms such as Superset and Proper Superset?

Answer 1

Indeed, $A$ is a superset of $B$ exactly when $B$ is a subset of $A$. $$A\supseteq B \iff B\subseteq A$$

Likewise $A$ is a proper superset of $B$ if and only if $B$ is a proper subset of $A$.

$$A\supsetneq B\iff B\subsetneq A$$

Answer 2

Notation and terminology can vary, but here at least the terminology is pretty much uniform. $A$ is a superset of $B$ if every element of $B$ is an element of $A.$ This does not preclude the case where $A=B$. $A$ is a proper superset of $B$ if it is a superset and it has at least one element that is not in $B$ (i.e. $A\ne B$). So any set is a superset of itself, but not a proper superset.

The first sentence on Mathworld is somewhere between misleading and wrong and you should ignore it (the linked page on proper superset is also poorly-worded in my opinion).

Additionally, be warned: their notation $A\subset B$ for proper subset/superset, despite the clear analogy with $<$ vs. $\le,$ is not quite standard. Often, $A\subset B$ is just used to denote subset/superset and $A\subsetneq B$ is reserved for proper subset/superset. The notation $\subseteq$ and $\subsetneq$ that Graham uses in his answer is what I prefer as well since it removes any ambiguity.