By using the set algebra , prove that , for any sets A and B
A u ( A' n B ) = A u B
Deduce , or using set algebra , show that
A u ( A' n B ) u [A' n (A u B') n {B u (B' n C}] = A u B u C
Hello mathaddict
Here's a proof - there might be a quicker one:
$\displaystyle A \cup (A'\cap B)\cup [A'\cap (A\cup B')\cap (B\cup (B'\cap C)]$
$\displaystyle =A\cup B\cup [A'\cap (A\cup B') \cap (B \cup C)]$, using the first result twice
$\displaystyle =A \cup B \cup[\{(A' \cap A) \cup (A'\cap B')\} \cap (B \cup C)]$, Distributive Law
$\displaystyle = A \cup B \cup [\{\oslash \cup (A' \cap B') \} \cap (B \cup C)]$, Complement Law
$\displaystyle = A \cup B \cup [(A' \cap B') \cap (B \cup C)]$, Identity Law
$\displaystyle = A \cup B \cup [(A \cup B)' \cap (B \cup C)]$, De Morgan's Law
$\displaystyle = [(A \cup B) \cup (A \cup B)'] \cap [(A \cup B) \cup (B \cup C)]$, Distributive Law
$\displaystyle = \boldsymbol{\text {U}} \cap [(A \cup B) \cup (B \cup C)]$, Complement Law
$\displaystyle = A \cup (B \cup B) \cup C$, Identity Law, Associative Law
$\displaystyle = A \cup B \cup C$, Idempotent Law
Grandad