NDA 2026 — Mathematics PYQ

To simplify this expression, we will analyze the components step-by-step using fundamental set theory identities.

Step 1: Simplify the first part: $((A\cap B)\cup (A-B))$

Using the property $A-B=A\cap B^c$:

$$(A\cap B)\cup (A\cap B^c)=A\cap (B\cup B^c)=A\cap U=A$$

So, the first part simplifies to $A$.

Step 2: Simplify the second part: $((A\cap B)\cup (B-A))$

Using the property $B-A=B\cap A^c$:

$$(A\cap B)\cup (B\cap A^c)=B\cap (A\cup A^c)=B\cap U=B$$

So, the second part simplifies to $B$.

Step 3: Combine and simplify the whole expression

Now, substitute these back into the original expression:

$$(A-B)\cup A$$

Since $(A-B)$ is essentially a subset of $A$ (because $A-B=A\cap B^c$, and $A\cap B^c\subseteq A$), the union of a set and its subset is simply the set itself:

$$A\cup A=A$$

Conclusion:

By breaking down the complex expression using set identities, we have determined that the final result is $A$.

Final Answer:

The expression is equal to $A$. The correct option is (b).