Question

Show that if AUB = A and AnB = A, then A= B.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

$\rm \: A\cup B = A - - - (1)$

and

$\rm \: A\cap B = A - - - (2)$

So, from equation (1) and (2), we concluded that

$\rm \: A\cup B = A\cap B$

Now, Let assume that

$\rm \: x \: \in \: A$

$\rm\implies \: x \: \in \: A\cup B$

$\rm\implies \: x \: \in \: A\cap B$

$\rm\implies \: x \: \in \: B$

$\rm\implies \:A \: \sub \: B - - - (3)$

Now, Consider

$\rm \: y \: \in \: B$

$\rm\implies \: y \: \in \: A\cup B$

$\rm\implies \: y \: \in \: A\cap B$

$\rm\implies \: y \: \in \: A$

$\rm\implies \:B \: \sub \: A - - - (4)$

From equation (3) and (4) , we concluded that

$\rm\implies \:A \: = \: B$

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ADDITIONAL INFORMATION

1. Commutative Law

$\rm \: A\cup B = B\cup A$

$\rm \: A\cap B = B\cap A$

2. Associative Law

$\rm \: (A\cup B)\cup C = A\cup (B\cup C) \:$

$\rm \: (A\cap B)\cap C = A\cap (B\cap C) \:$

3. Distributive Law

$\rm \: A\cup (B\cap C) = (A\cup B)\cap (A\cup B)$

$\rm \: A\cap (B\cup C) = (A\cap B)\cup (A\cap B)$

4. De Morgan's Law

$\rm \: (A\cup B)' = A'\cap B'$

$\rm \: (A\cap B)' = A'\cup B'$