GATE CS 2015 Set 3 Q23 — Sets, Relations and Functions

Let's evaluate each option to find the correct statement:

• Option A: False. If $X = \{1\}$, then $|X| = 1$ and $|X'| = |S \setminus X| = 5$. Thus, $|X| \neq |X'|$.
• Option B: False. If $|X| = 5$ and $|Y| = 5$, then by the principle of inclusion-exclusion, $|X \cap Y| = |X| + |Y| - |X \cup Y| \ge 5 + 5 - 6 = 4$. Thus, $X \cap Y$ cannot be empty.
• Option C: False. If we choose $X = \{1, 2\}$ and $Y = \{3, 4, 5\}$, then $X \setminus Y = \{1, 2\} \neq \emptyset$.
• Option D: True. Using standard set theory identities:
  $$Y' \setminus X' = Y' \cap (X')' = Y' \cap X = X \cap Y' = X \setminus Y$$
  Since this identity holds for any sets $X$ and $Y$, the statement is universally true.