NIMCET 2021 — Mathematics PYQ

According to De Morgan's Law, the complement of the union of two sets is the intersection of their complements:

Step 2: Substitute this back into the original expression.

The expression becomes:

Step 3: Apply the Associative and Commutative Laws.

Since all the operations are intersections ($\cap$), we can rearrange the terms:

Step 4: Use the Complement Law.

The intersection of a set and its own complement is always an empty set:

Also, the intersection of a set with itself is the set itself:

Step 5: Final simplification.

Any set intersected with an empty set results in an empty set:

Final Answer:

The value of $X\cap Y^{\prime }\cap (X\cup Y{)}^{\prime }$ is $\varphi$ (the null set).