Set theory notation

UNDER CONSTRUCTION

Set theory notation is useful when talking about probabilities

If \(x\) is a member or element of \(A\), then we write \(x \in A\). Further:

  • Sets and elements:
    • \(x \notin A \): \(x\) is not an element of \( A \)
    • \(A = \{x,y,z\}\) if \(A\) is the set only with elements \(x, y, z\), e.g. \(A=\{1,2,3\}\), \(1\in A\) and \(4 \notin A\)
    • \(A = \{x; S(x)\} \): a set \(A\) with elements \(x\) for which statement \(S(x)\) hold
    • \( \emptyset= \{x;x \neq x\} \) is the null set (the set with no elements)
    • \( x \notin \emptyset \) for all \( x\)
    • \( A \subset B \): \( A \) is a subset of \( B\), e.g. \(\{1,2\} \subset \{1,2,4\}\), so if \(x \in A \) then also \( x \in B \), but not necessarily vice versa
    • \( A \supset B \): \( A \) is a superset of \( B\), e.g. \(\{1,2,4\} \supset \{1,2\}\), so if \(x \in B \) then also \( x \in A \)
    • \( \emptyset \subset A \), \(A \subset A\), \( A \supset A\) for all \( A \)
  • Union
    • \(A \cup B = \{ x; x \in A \) and/or \( x \in B\}\)
    • the union of sets \( A \) and \( B\)
    • e.g. \(\{1,2,4\} \cup \{1,3\} = \{1,2,3,4 \}\)
  • Intersection
    • \(AB = A \cap B = \{x; x \in A \) and \( x \in B\} \)
    • the intersection of \( A \) and \( B \)
    • e.g. \(\{1,2,4\} \cap \{1,3\} = \{1 \}\)
  • Set difference
    • \( A \setminus B = \{x; x \in A \) but \(x \notin B\}\)
    • the difference set \(A\) less \(B\)
    • e.g. \(\{1,2,4\} \setminus \{1\} = \{2,4 \}\)
  • Sequences
    • \( (A_n) \) is a sequence of sets \( A_1, A_2, A_3,... , A_n \)