Subsets and Proper Subsets

\({A \subset B, A}\) is a proper subset of \({B}\) if all elements of \({A}\) are elements of \({B}\) and \({B \neq A}\).

Consider the set \({B = \{2, 4, 6, 8, 10\}}\) and \({A = \{4, 8\}}\)

• Every element of \({A}\) is an element of \({B}\) BUT \({B \neq A}\).
\(\:\) \({\therefore A}\) is a proper subset of \({B}\).

Note:
1. The empty set \({\varnothing}\) is a (proper) subset of any non-empty set;
2. Any set is a subset of itself.