1.2 Subsets

Subset: \(A\) is a subset of \(B\), written \(A \subseteq B \), means every element of \(A\) is also an element of \(B\).

Notation: \(C \nsubseteq D\) means there is an element of \(C\) which does not lie in \(D\).

Note: Let \(A\) be a set.

\[\emptyset \subseteq A\]

Familiar sets in \(\mathbb{R}\), intervals: Nine types:

  • \((a, b) = \{ x \in \mathbb{R} : a < x < b \}\)
  • \((a, b] = \{ x \in \mathbb{R} : a < x \leq b \}\)
  • \(\vdots\) (fill in the rest)

Note: \((a, b) \neq \{ a, b \}\)

Equality: What does \(A = B\) mean?

\[ A \subseteq B, B \subseteq A \]

Venn Diagrams are useful set visualizations!

Proper Subset: \(A\) is a proper subset if \(A \subseteq B\) and \(A \neq B\), written \(A \subset B\).

Power Set: Given a set \(A\), the set of all subsets of \(A\) is called the power set of \(A\).

\[ \text{power set}(A) = \mathbb{P}(A) \]

Example: \(A = \{ a, b \}\) \[ \mathbb{P}(A) = \{ \emptyset, \{ a \}, \{ b \}, \{ a, b \} \} \] Observe: \[ |\mathbb{P}(A)| = 2^{|A|} \]

Fact: If \(A\) is a finite set, then \(\mathbb{P}(A) = 2^{|A|}\). (we will prove this later)