Subsets

Definition

Suppose that \(A\) and \(B\) are sets.

  • If every element of \(A\) is also an element of \(B\), then we say that \(A\) is a subset of \(B\). This can be written using the subset symbol \(A\subseteq B\).

  • If at least one element of \(A\) is not an element of \(B\), then \(A\) is not a subset of \(B\). This can be written \(A\not\subseteq B\).

Example

  • \(\{2,3,7\}\subseteq\{2,3,4,5,6,7\}\)
  • \(\{2,3,11\}\not\subseteq\{2,3,4,5,6,7\}\)

Example

  • \(\mathbb{N}\subseteq\mathbb{Z}\)
  • \(\mathbb{Z}\subseteq\mathbb{Q}\)

Example

  • \(\mathbb{R}\times\mathbb{N}\subseteq\mathbb{R}\times\mathbb{R}\)

Example

  • \(\mathbb{N}\times \mathbb{R}\not\subseteq\mathbb{R}\times\mathbb{N}\)

Example

  • For any set \(A\), \(A\subseteq A\).

The Empty Set

  • The empty set is a subset of every set.