Prove That If A ⊂ B Then A ∩ B = A

Solution

Given:

\[ A\subset B \]

This means every element of \( A \) is also an element of \( B \).

Let \( x\in A \).

Since \( A\subset B \), therefore:

\[ x\in B \]

Hence \( x \) belongs to both \( A \) and \( B \).

Therefore,

\[ x\in A\cap B \]

Thus every element of \( A \) belongs to \( A\cap B \), so:

\[ A\subset A\cap B \]

Also, every element of \( A\cap B \) belongs to \( A \).

Therefore,

\[ A\cap B\subset A \]

Since

\[ A\subset A\cap B \quad \text{and} \quad A\cap B\subset A \]

we get:

\[ A\cap B=A \]

Hence proved.