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 $$A_1, A_2, A_3,... , A_n$$