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

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- Dec 11th 2008, 06:52 AMmathaddictAnother sets prove
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 - Dec 11th 2008, 07:04 AMtester85
- Dec 12th 2008, 01:03 AMmathaddict
- Dec 12th 2008, 06:28 AMGrandadProofs using the Laws of Sets
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