Subsets

Every subset is a smaller set living within a larger one—like a slice of the whole pie! Understanding subsets helps us explore relationships and structure within sets. 🥧

Subset

Suppose \(A\) and \(B\) are sets.
If every element of \(A\) is also an element of \(B\), then we say \(A\) is a subset of \(B\), denoted \(A \subseteq B\).
We write \(A \not\subseteq B\) if \(A\) is not a subset of \(B\), that is, if it is not true that every element of \(A\) is also an element of \(B\). Thus \(A \not\subseteq B\) means that there is at least one element of \(A\) that is not an element of \(B\).

\[ \mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{R} \]

\[ A = \{1,2\}, B = \{2,3,4\} \] \[ A \not\subseteq B \]

Exercise

List all the subsets of the following sets:

  1. \(\{1,2,3\}\)
  2. \(\{1,\{2,3\}\}\)
  3. \(\{\mathbb{N}, \mathbb{Z}, \mathbb{R}\}\)
  1. \(\{\}, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}\)
  2. \(\{\}, \{1\}, \{\{2,3\}\}, \{1,\{2,3\}\}\)
  3. \(\{\}, \{\mathbb{N}\}, \{\mathbb{Z}\}, \{\mathbb{R}\}, \{\mathbb{N},\mathbb{Z}\}, \{\mathbb{N},\mathbb{R}\}, \{\mathbb{Z},\mathbb{R}\}, \{\mathbb{N},\mathbb{Z},\mathbb{R}\}\)