
# 3.4: Summary


• Important definitions:
• set
• element
• subset
• proper subset
• predicate
• union
• intersection
• set difference
• complement
• disjoint
• pairwise-disjoint
• power set
• A set is unordered and without repetition.
• $$\emptyset$$ and $$A$$ are subsets of $$A$$.
• $$A = B$$ if and only if we have both $$A \subset B$$ and $$B \subset A$$.
• For our purposes, predicates usually have only one or two variables.
• If a predicate has two variables, the order of the variables is important.
• Venn diagrams are a tool for illustrating set operations.
• $$\#\mathcal{P}(A) = 2^{\#A}$$
• Notation:
• $$\{ \ \ \ \}$$
• $$\in$$, $$\notin$$
• $$\emptyset$$ (empty set)
• $$\#A$$
• $$A \subset B$$, $$A \not\subset B$$, $$A \supset B$$
• $$P(x)$$, $$x \mathrel{Q} y$$ (predicates)
• $$\{\, a \in A \mid P(a) \,\}$$
• $$\mathcal{U}$$ (universe of discourse)
• $$\mathbb{N}$$, $$\mathbb{N}^+$$, $$\mathbb{Z}$$, $$\mathbb{Q}$$, $$\mathbb{R}$$
• $$A \cup B$$
• $$A \cap B$$
• $$A \setminus B$$
• $$\bar{A}$$
• $$\mathcal{P}(A)$$