Discrete-math 4

Cartesian product

\begin{matrix} A \times B = {(a, b) ∣ a \in A, b \in B} \end{matrix}

\begin{matrix} Prove: A \times (B \cap C) = (A \times B) \cap (A \times C) \\ Proof: \\ (a, b) \in A \times (B \cap C) ⟺ a \in A, b \in B \cap C ⟺ a \in A \land b \in B \land b \in C \\ ⟺ a \in A \land b \in B \land a \in A \land b \in C ⟺ (a \in A \land b \in B) \land (a \in A \land b \in C) \\ ⟺ (a, b) \in (A \times B) \land (a, b) \in (A \times C) ⟺ (a, b) \in (A \times B) \cap (A \times C) \end{matrix}

R over A, B : R \subseteq (A \times B)

\forall a \in A : (a, a) \in R

\forall a, b \in A : (a, b) \in R ⟹ (b, a) \in R

\forall a, b, c \in A : ((a, b) \in R \land (b, c) \in R) ⟹ (a, c) \in R