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,

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

The RHS is equivalent to

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

Which is equivalent to

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

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

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

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

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

The RHS is equivalent to

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

Which is equivalent to

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

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

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

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

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

The RHS is equivalent to

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

Which is equivalent to

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

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