Tao Analysis I - 3.5.4

Exercise 3.5.4

Let $A$, $B$, $C$ be sets. Show that $A \times (B \cup C)=(A \times B) \cup (A \times C)$, that $A \times (B \cap C)=(A \times B) \cap (A \times C)$, and that $A \times (B \setminus C)=(A \times B) \setminus (A \times C)$.

Show $A \times (B \cup C)=(A \times B) \cup (A \times C)$

Let's write out what $A \times (B \cup C)$ means,

$$(a,d) \in A \times (B \cup C) \iff (a,d) \in \{ (x,y) : x \in A,  y \in B \cup C\}$$

The RHS is equivalent to

$$(a,d) \in \{ (x,y) : x \in A,  y \in B\} \lor (a,d) \in \{ (x,y) : x \in A,  y \in C\}$$

Which is equivalent to

$$(a,d) \in A \times (B \cup C) \iff (a,d) \in (A \times B) \cup (A \times C)$$

Thus, we have shown $A \times (B \cup C)=(A \times B) \cup (A \times C)$. $\square$

Show $A \times (B \cap C)=(A \times B) \cap (A \times C)$

Let's write out what $A \times (B \cap C)$ means

$$(a,d) \in A \times (B \cap C) \iff (a,d) \in \{ (x,y) : x \in A,  y \in B \cap C\}$$

The RHS is equivalent to

$$(a,d) \in \{ (x,y) : x \in A,  y \in B\} \land (a,d) \in \{ (x,y) : x \in A,  y \in C\}$$

Which is equivalent to

$$(a,d) \in A \times (B \cap C) \iff (a,d) \in (A \times B) \cap (A \times C)$$

Thus, we have shown $A \times (B \cap C)=(A \times B) \cap (A \times C)$. $\square$

Show $A \times (B \setminus C)=(A \times B) \setminus (A \times C)$

Let's write out what $A \times (B \setminus C)$ means

$$(a,d) \in A \times (B \setminus C) \iff (a,d) \in \{ (x,y) : x \in A,  y \in B \setminus C\}$$

The RHS is equivalent to

$$(a,d) \in \{ (x,y) : x \in A,  y \in B\} \land (a,d) \notin \{ (x,y) : x \in A,  y \in C\}$$

Which is equivalent to

$$(a,d) \in A \times (B \setminus C) \iff (a,d) \in (A \times B) \setminus (A \times C)$$

Thus, we have shown $A \times (B \setminus C)=(A \times B) \setminus (A \times C)$. $\square$