GATE CS 2008 Q2 — Sets, Relations and Functions

We can simplify the given set expression step-by-step using standard set identities:

1. Simplify the first two terms:

   Using the distributive law, we can factor out $(Q \cap R)$:

   $$(P \cap Q \cap R) \cup (P^c \cap Q \cap R) = (P \cup P^c) \cap (Q \cap R)$$

   Since $P \cup P^c = U$ (the universal set):

   $$U \cap (Q \cap R) = Q \cap R$$
2. Substitute back into the full expression:

   The expression now becomes:

   $$(Q \cap R) \cup Q^c \cup R^c$$
3. Apply De Morgan's Law:

   Recall that $Q^c \cup R^c = (Q \cap R)^c$. Substituting this in gives:

   $$(Q \cap R) \cup (Q \cap R)^c$$
4. Final simplification:

   For any set $A$, we have $A \cup A^c = U$. Letting $A = Q \cap R$:

   $$(Q \cap R) \cup (Q \cap R)^c = U$$

Thus, the expression simplifies to the universal set $U$.