Question

Consider the following statements:
$S_1$: There exist infinite sets $A,B,C$ such that $A\cap (B\cap C)$ is finite.
$S_2$: There exist two irrational numbers $x$ and $y$ such that $(x+y)$ is rational.
Which of the following is true about $S_1$ and $S_2$ ?

• A. Only $S_1$ is correct
• B. Only $S_2$ is correct
• C. Both $S_1$ and $S_2$ are correct
• D. None of $S_1$ and $S_2$ is correct

Answer 1

The correct option is C Both $S_1$ and $S_2$ are correct

$S_1:$ Let $A=$ set of integers
$B=$ set of odd integers
$C=$ set of even integers
$A\cap (B\cap C)=\varphi$ and $\varphi$ is finite set.
Therefore, $S_1$ is true.

$S_2:$ Let two irrational number $x$ and $y$ are respectively $(1+\sqrt{2})$ and $(1-\sqrt{2})$
so $x+y=1+\sqrt{2}+1-\sqrt{2}$
$=2$ which is rational number
Therefore, $S_2$ is true. Since both $S_1$ and $S_2$ are true, option (c) is true.