Using properties of sets show that:
$(i) A \cup (A \cap B) = A$
$(i i) A \cap (A \cup B) = A$

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Solution

$(i) A \cup (A \cap B) = A$
L.H.S $= A \cup (A \cap B)$
$= (A \cup A) \cap (A \cup B) {Using distributive law}$
$= A \cap (A \cup B)$
$= A$
= R.H.S
Thus, $A \cup (A \cap B) = A$
Hence proved.

$(i i) A \cap (A \cup B) = A$
L.H.S. $= A \cap (A \cup B)$
$= (A \cap A) \cup (A \cap B) {Using distributive law}$
$= A \cup (A \cap B)$
$= A$
= R.H.S
Thus, $A \cap (A \cup B) = A$
Hence proved.

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