A B means every element of A belongs to B; equality is allowed. A B means a proper subset: every element belongs to B, but A!= B. A nested Venn region represents this containment. **Statement** | **How to check** A B | Find an element of A outside B. A B | A is not a proper subset: either containment fails or A=B. Example For A=4,8 and B=2,4,6,8,10, A B. Also A A, but A A. Since 2 in B but 2 A, B A. Use in between an element and a set; use between sets. Here 4 in A, 4 A, and 4 A. The empty set is a subset of every set.

\(A\subseteq B\) means every element of \(A\) belongs to \(B\); equality is allowed. \(A\subset B\) means a proper subset: every element belongs to \(B\), but \(A\ne B\). A nested Venn region represents this containment.

For \(A=\{4,8\}\) and \(B=\{2,4,6,8,10\}\), \(A\subset B\). Also \(A\subseteq A\), but \(A\not\subset A\). Since \(2\in B\) but \(2\notin A\), \(B\nsubseteq A\).

Use \(\in\) between an element and a set; use \(\subseteq\) between sets. Here \(4\in A\), \(\{4\}\subset A\), and \(\{4\}\notin A\). The empty set is a subset of every set.

Statement	How to check
\(A\nsubseteq B\)	Find an element of \(A\) outside \(B\).
\(A\not\subset B\)	\(A\) is not a proper subset: either containment fails or \(A=B\).

Subsets and proper subsets

Subsets and proper subsets