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 $U$:

- The set of all animals is the superset of all dogs.
- The set of $A=\{\text{Italy,Georgia,Brasil}\}$ is a subset of the set $C=\{x:\, x \text{ is a country}\}.$
- The set of $\mathbb Z$ of all integers is a subset of the set $\mathbb Q$ of all rational numbers.
- A line on a plane is a subset of this plane.

| | | | | created: 2014-03-22 15:59:20 | modified: 2019-09-07 15:17:24 | by: *bookofproofs* | references: [979], [7838]

[7838] **Kohar, Richard**: “Basic Discrete Mathematics, Logic, Set Theory & Probability”, World Scientific, 2016

[979] **Reinhardt F., Soeder H.**: “dtv-Atlas zur Mathematik”, Deutsche Taschenbuch Verlag, 1994, 10