Cartesian Product Distributes over Set Difference

Theorem

Cartesian product is distributive over set difference:

$(1): \quad S \times \paren {T_1 \setminus T_2} = \paren {S \times T_1} \setminus \paren {S \times T_2}$

$(2): \quad \paren {T_1 \setminus T_2} \times S = \paren {T_1 \times S} \setminus \paren {T_2 \times S}$

Proof

\(\ds \)

\(\)

\(\ds \tuple {x, y} \in S \times \paren {T_1 \setminus T_2}\)

\(\ds \)

\(\leadstoandfrom\)

\(\ds \paren {x \in S} \land \paren {y \in \paren {T_1 \setminus T_2} }\)

Definition of Cartesian Product

\(\ds \)

\(\leadstoandfrom\)

\(\ds \paren {x \in S} \land \paren {y \in T_1} \land \paren {y \notin T_2}\)

Definition of Set Difference

\(\ds \)

\(\leadstoandfrom\)

\(\ds \paren {\tuple {x, y} \in S \times T_1} \land \paren {\tuple {x, y} \notin S \times T_2}\)

Definition of Cartesian Product

\(\ds \)

\(\leadstoandfrom\)

\(\ds \tuple {x, y} \in \paren {S \times T_1} \setminus \paren {S \times T_2}\)

Definition of Set Difference

\(\ds \)

\(\)

\(\ds \tuple {x, y} \in \paren {T_1 \setminus T_2} \times S\)

\(\ds \)

\(\leadstoandfrom\)

\(\ds \paren {x \in \paren {T_1 \setminus T_2} } \land \paren {y \in S}\)

Definition of Cartesian Product

\(\ds \)

\(\leadstoandfrom\)

\(\ds \paren {x \in T_1} \land \paren {x \notin T_2} \land \paren {y \in S}\)

Definition of Set Difference

\(\ds \)

\(\leadstoandfrom\)

\(\ds \paren {\tuple {x, y} \in T_1 \times S} \land \paren {\tuple {x, y} \notin T_2 \times S}\)

Definition of Cartesian Product

\(\ds \)

\(\leadstoandfrom\)

\(\ds \tuple {x, y} \in \paren {T_1 \times S} \setminus \paren {T_2 \times S}\)

Definition of Set Difference

$\blacksquare$

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 6$: Ordered Pairs: Exercise $\text{(iii)}$
• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts: Exercise $1.2 \ \text{(o)}$