GATE CS 2005 Q8 — Sets, Relations and Functions

We can simplify both set expressions using the identity $P - Q = P \cap Q^c$ (where $Q^c$ denotes the complement of set $Q$).

1. Simplifying $X$:

$$\begin{aligned} X &= (A - B) - C \\ &= (A \cap B^c) \cap C^c \\ &= A \cap B^c \cap C^c \end{aligned}$$

2. Simplifying $Y$:

$$\begin{aligned} Y &= (A - C) - (B - C) \\ &= (A \cap C^c) \cap (B \cap C^c)^c \end{aligned}$$

Applying De Morgan's Law to the second term, $(B \cap C^c)^c = B^c \cup (C^c)^c = B^c \cup C$:

$$\begin{aligned} Y &= (A \cap C^c) \cap (B^c \cup C) \\ &= (A \cap C^c \cap B^c) \cup (A \cap C^c \cap C) \end{aligned}$$

Since $C^c \cap C = \emptyset$, the second term becomes empty:

$$\begin{aligned} Y &= (A \cap B^c \cap C^c) \cup \emptyset \\ &= A \cap B^c \cap C^c \end{aligned}$$

Since both $X$ and $Y$ simplify to $A \cap B^c \cap C^c$, we have:

$$X = Y$$

Thus, the correct option is A.