1 sets

Expressed with capital letters or list of elements between braces:

A = \{ a_1, a_2, \cdots, a_n \} where n is the size of the set

Two sets are considered equal when all the elements are the same

We consider subsets of some universal X, A, B, \cdots \in 2^X = the set of all subsets of X:

let: X = \{ a, b, c \} then 2^X = \{ \emptyset, \{ a \}, \{ b \}, \{ c \}, \{ b, c \}, \{ a, b \}, \{ a, c \}, \{ a, b, c \} \}

|2^X| = 2^{|X|}

2^X = p(X), also: p_k(X) is a set of sets of length k

1.0.0.1 declarations

• A \subseteq B: Every element of A is an element of B
• A \subsetneq B: Every element of A is an element of B but not equal to each other
• a \in B: a is an element of B

1.0.0.2 De Morgan’s law

\bigcup\limits_{t \in T}(\mathbb{R} - A_t) = \mathbb{R} - \bigcap\limits_{t \in T} A_t

1.0.0.3 operations on sets

Assumption: x \in X

• union: A \cup B = x \in A \lor x \in B
• intersection: A \cap B = x \in A \land x \in B
• difference: A - B = x \in A \land x \notin B¹
• symmetric difference: A \div B = (A - B) \cup (B - A)
• complement: A' = x \notin A
• cartesian product: \{1, 2\} \times \{3, 4, 5\} = \{(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)\}
• union of sets: \bigcup\limits_{i=1}^{n} S_i
• intersection of sets: \bigcap\limits_{i=1}^{n} S_i