Let \(X\) be a set. A set \(A\) is called a subset of \(X\) and denoted by \(A\subseteq X\), if each element of \(A\) is also an element of \(X\). Equivalently, X is a superset of \(A\), denoted as \(X\supseteq A\).
If, in addition, \(A\neq X\), then \(A\) is called a proper subset of \(X\) and denoted by \(A\subset X\). Equivalently, X is then a proper superset of \(A\), and denoted as \(X\supset A\).
We can draw supersets and subsets as Venn diagrams. In the following figure, $B$ is a subset of $A$ and $A$ is a subset of the universal set $U$:
| | | | | created: 2014-03-22 15:59:20 | modified: 2019-09-07 15:17:24 | by: bookofproofs | references: , 
 Kohar, Richard: “Basic Discrete Mathematics, Logic, Set Theory & Probability”, World Scientific, 2016
 Reinhardt F., Soeder H.: “dtv-Atlas zur Mathematik”, Deutsche Taschenbuch Verlag, 1994, 10