Using properties of sets show that
Question:

Using properties of sets show that

(i) $A \cup(A \cap B)=A$ (ii) $A \cap(A \cup B)=A$.

Solution:

(i) To show: $A \cup(A \cap B)=A$

We know that

$A \subset A$

$A \cap B \subset A$

$\therefore A \cup(A \cap B) \subset A \ldots(1)$

Also, $A \subset A \cup(A \cap B) \ldots$ (2)

$\therefore$ From $(1)$ and $(2), A \cup(A \cap B)=A$

(ii) To show: $A \cap(A \cup B)=A$

$A \cap(A \cup B)=(A \cap A) \cup(A \cap B)$

$=A \cup(A \cap B)$

$=A\{$ from $(1)\}$