3.4: Summary
- Page ID
- 23894
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- 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)
- \(\mathbb{N}\), \(\mathbb{N}^+\), \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{R}\)
- \(A \cup B\)
- \(A \cap B\)
- \(A \setminus B\)
- \(\bar{A}\)
- \(\mathcal{P}(A)\)