Prove That A ⊂ B ⇒ C − B ⊂ C − A

Solution

Given:

\[ A\subset B \]

Let \( x\in C-B \).

Then,

\[ x\in C \quad \text{and} \quad x\notin B \]

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

Therefore, if \( x\notin B \), then \( x\notin A \).

Thus,

\[ x\in C \quad \text{and} \quad x\notin A \]

Hence,

\[ x\in C-A \]

Therefore every element of \( C-B \) is also an element of \( C-A \).

So,

\[ C-B\subset C-A \]

Hence proved.