GATE CS 2001 Q2.2 — Sets, Relations and Functions

We analyze the two statements individually:

Statement $S_1$: There exist infinite sets $A$, $B$, $C$ such that $A \cap (B \cup C)$ is finite.

• Let $A$ be the set of all even positive integers: $A = \{2, 4, 6, 8, \dots\}$ (infinite set).
• Let $B$ be the set of all odd positive integers: $B = \{1, 3, 5, 7, \dots\}$ (infinite set).
• Let $C$ be the set of all odd positive integers: $C = \{1, 3, 5, 7, \dots\}$ (infinite set).

Here, $B \cup C = \{1, 3, 5, 7, \dots\}$.

The intersection is: $$A \cap (B \cup C) = \emptyset$$ Since the empty set $\emptyset$ is finite (cardinality $0$), such infinite sets $A$, $B$, and $C$ indeed exist. Thus, $S_1$ is correct.

Statement $S_2$: There exist two irrational numbers $x$ and $y$ such that $(x+y)$ is rational.

• Let $x = \sqrt{2}$, which is an irrational number.
• Let $y = -\sqrt{2}$, which is also an irrational number.

Their sum is: $$x + y = \sqrt{2} + (-\sqrt{2}) = 0$$ Since $0$ is a rational number, we have found two irrational numbers whose sum is rational. Thus, $S_2$ is correct.

Since both statements $S_1$ and $S_2$ are correct, the correct option is C.