Prove That A ∪ (A ∩ B) = A

Solution

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

Then either

\[ x\in A \]

or

\[ x\in A\cap B \]

If \( x\in A\cap B \), then by definition of intersection,

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

Therefore in both cases,

\[ x\in A \]

Hence,

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

Now let \( x\in A \).

Since every element of \( A \) belongs to the union

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

therefore,

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

Hence,

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

Since

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

and

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

therefore,

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

Hence proved.