Primary reference: Chapter 1 (Sets) and Chapter 7 (Proofs Involving Sets) of Book of Proof by Richard Hammack

richardhammack.github.io

Content covered in this unit includes:

• Describing a set with set-builder notation or by listing its elements

• The idea that two sets are equal if and only if they contain the same elements

• The cardinality (or size) of a set, and the meaning of the terms finite and infinite

• The sets $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$

• The empty set $\varnothing = \{\}$, which contains no elements

• Intervals in the set of real numbers (open, closed, half-open, or infinite)

• The union and intersection of two or more sets, and the difference of two sets

• The complement of a set relative to some universal set

• The use of Venn diagrams to visually represent operations on sets (union, intersection, difference, complement)

• The definition of a subset of a set, and the power set of a set

• The definition of an ordered pair, and the Cartesian product of two sets

• The strategy for proving that something is an element of a set:

  <aside>

  Proposition. $a \in \{x : P(x)\}$.

  Proof. Suppose $a$.

  …

  Therefore $P(a)$.

  </aside>

  Or:

  <aside>

  Proposition. $a \in \{x \in X : P(x)\}$.

  Proof. Suppose $a$.

  …

  Thus $x \in X$.

  …

  Therefore $P(a)$.

  </aside>

• The strategy for proving that a set is a subset of another set:

  <aside>

  Proposition. $A \subseteq B$.

  Proof. Suppose $a \in A$.

  …

  Therefore $a \in B$. We conclude that $A \subseteq B$.

  </aside>

• The strategy for proving that two sets are equal:

  <aside>

  Proposition. $A = B$.

  Proof. We will first show that $A \subseteq B$. So, suppose $a \in A$.

  …

  Therefore $a \in B$. We conclude that $A \subseteq B$. Next we will show that $B \subseteq A$. So, suppose $b \in B$.

  …

  Therefore $b \in A$. We conclude that $B \subseteq A$, and thus that $A = B$.

  </aside>