Using properties of sets, show that: (i) A ∪ (A ∩ B) = A (ii) A ∩ (A ∪ B) = A.

Home » Using properties of sets, show that: (i) A ∪ (A ∩ B) = A (ii) A ∩ (A ∪ B) = A.

Using properties of sets, show that: (i) A ∪ (A ∩ B) = A (ii) A ∩ (A ∪ B) = A.

Solution:

(i) We need to show: $A \cup(A \cap B)=A$

As it is known that,

$\begin{array}{l} A \subset A \\ A \cap B \subset A \\ \therefore A \cup(A \cap B) \subset A..........(i) \end{array}$

Also, as per the question,

We have:

$A \subset A \cup(A \cap B)$ ………….(ii)

As a result, from eq. (i) and (ii)

We have:

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

(ii) We need to show,

$\begin{array}{l} A \cap(A \cup B)=A \\ A \cap(A \cup B)=(A \cap A) \cup(A \cap B) \\ =A \cup(A \cap B) \\ =A \end{array}$

i

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