GATE CS 1993 Q8.3 — Sets, Relations and Functions

We are given that $S$ is an infinite set and is the union of $n$ sets:

Let us analyze the properties of the sets $S_i$:

• Proof for Option C: Suppose none of the sets $S_i$ are infinite. This means every $S_i$ (for $1 \le i \le n$) is a finite set. Since the union of a finite number of finite sets is always finite, the union $S = \bigcup_{i=1}^n S_i$ must also be finite. However, this contradicts the given fact that $S$ is an infinite set. Therefore, our assumption is false, and at least one of the sets $S_i$ must be infinite.

We can eliminate the other options with simple counterexamples:

• Option A is incorrect: Let $S = \mathbb{Z}$ (the set of all integers, which is infinite). Let $S_1 = \text{even integers}$ and $S_2 = \text{odd integers}$. Both $S_1$ and $S_2$ are infinite, so neither is finite.
• Option B is incorrect: Let $S = \mathbb{Z}$. Let $S_1 = \{1\}$, $S_2 = \{2\}$, and $S_3 = \mathbb{Z} \setminus \{1, 2\}$. Here, two sets ($S_1$ and $S_2$) are finite.
• Option D is incorrect: In the counterexample for Option A, both $S_1$ and $S_2$ are infinite, showing that more than one set can be infinite.