Prove That A ∩ (A ∪ B) = A

Solution

Let \( x\in A\cap(A\cup B) \).

Then,

\[ x\in A \]

and

\[ x\in A\cup B \]

Since \( x\in A \), therefore every element of

\[ A\cap(A\cup B) \]

belongs to \( A \).

Hence,

\[ A\cap(A\cup B)\subset A \]

Now let \( x\in A \).

Since every element of \( A \) belongs to \( A\cup B \),

\[ x\in A\cup B \]

Therefore,

\[ x\in A \quad \text{and} \quad x\in A\cup B \]

Hence,

\[ x\in A\cap(A\cup B) \]

Thus,

\[ A\subset A\cap(A\cup B) \]

Since

\[ A\cap(A\cup B)\subset A \]

and

\[ A\subset A\cap(A\cup B) \]

therefore,

\[ A\cap(A\cup B)=A \]

Hence proved.