Using set algebra, show that: A - [ B - (A ∩ B) ] = A

To solve this using set algebra, we can use the properties of set operations.

We will start by expanding the expression:

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

Using the definition of set difference, we have:

A \cap (B \cap (A \cap B)^\complement )

Since A \cap B \subseteq A and B - (A \cap B) means elements in B but not in A \cap B , we can rewrite as:

A \cap (B \cap A^\complement)

Applying the distributive property of intersection over union, we get:

A \cap (B \cap A) \cap (B \cap A^\complement)

Applying the idempotent law, where A \cap A = A and B \cap B = B , we simplify further:

A \cap B \cap A \cap B^\complement

Given that A \cap B \subseteq A , we are left with:

A

Therefore,

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

\boxed{A}

In conclusion, we have shown using set algebra that the given expression is equal to set A.