Set Theory

2026-01-14

Set Theory

Definitions

Universal Set

The Universal set \(u\) is the set of all elements that can be chosen from. It changes for every problem.

Set operations

• \(A \cup B\) or union means the set of all elements in \(A\) , or \(B\) , or both.
• \(A \cap B\) or intersection means the set of all elements in both A and \(B\) .
• \(A - B\) or \(A \backslash B\) means set difference. remove all elements from \(B\) that are in \(A\) . \(A - B = A \cap B^{\complement}\) .
• \(A^{\complement}\) or \(\overline{A}\) means the set of everything not in \(A\) .

Partitions

\(A\) and \(B\) is disjoint if \(A \cap B = \emptyset\) .

\(A_{1}, A_{2}, ..., A_{n}\) are mutually disjoint if \(A_{i} \cap A_{j} = \emptyset\) for all \(i \neq j\) .

A set \(\{A_{1}, A_{2}, ..., A_{n}\}\) is a partition of a set \(A\) if \(A_{1}, A_{2}, ..., A_{n}\) are mutually disjoint, and \(A = A_{1} \cup A_{2} \cup ... \cup A_{n}\) . \(A_{n}\) may not be \(\emptyset\) .

Power Set

\(\wp(A)\) is the set of all subsets of A.

\(\lvert \wp(A) \rvert = 2^{\lvert A \rvert}\)

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