Cardinality and Set Theory

Question

(|x| = cardinal # of x for clarification)

let A,B be two finite sets, show that $|A \cup B| = |A| + |B| -|A\cap B|$

Proof: let $x\in A\cup B$

$x \in (A -A\cap B) + (B- A\cap B) + (A \cap B)$

let |A|=a, |B|=b, $|A\cap B|$=c

then rewrite it as

(a-c) + (b-c) + c $\implies$ a+b-2c+c $\implies$a+b-c

rewritten as

|A| + |B| - $|A\cap B|$

Answer 1

$x\in A\cup B \Leftrightarrow x \in (A -A\cap B) \cup (B- A\cap B) \cup (A \cap B)$

So :

$$|A \cup B|=|A-A \cap B|+|B-A \cap B|+|A \cap B|=|A|-|A \cap B|+|B|-|A \cap B|+|A \cap B|=|A|+|B|-|A \cap B|$$