53 votes 53 votes Consider the following statements: $S_1:$ There exists infinite sets $A$, $B$, $C$ such that $A \cap (B \cup C)$ is finite. $S_2:$ There exists two irrational numbers $x$ and y such that $(x+y)$ is rational. Which of the following is true about $S_1$ and $S_2$? Only $S_1$ is correct Only $S_2$ is correct Both $S_1$ and $S_2$ are correct None of $S_1$ and $S_2$ is correct Set Theory & Algebra gatecse-2001 set-theory&algebra normal set-theory + – Kathleen asked Sep 14, 2014 edited Mar 26, 2021 by soujanyareddy13 Kathleen 8.8k views answer comment Share Follow See all 4 Comments See all 4 4 Comments reply ESHA_SHARMA commented Oct 9, 2017 reply Follow Share CAN THERE BE A POSSIBILITY IN S1 THAT A,B,C ARE NOT MUTUALLY EXCLUSIVE SETS(BECAUSE ONLY INFINITE IS MENTIONED AND NOTHING HAS BEEN SAID ABOUT THEIR RELATION WITH EACH OTHER)-THEN S1 CAN BE EITHER TRUE AND EITHER FALSE.? 0 votes 0 votes Rupendra Choudhary commented Dec 22, 2017 reply Follow Share Hello Esha Yes! it can be. although in question there is 'There exist' which means if we find a single such combination of A,B,C that result finite outcome then statement will be true. even if he says A,B,C are not disjoint sets then still we can find such arrangement that will make that statement true like A={a*} B={ab*} and C={aab*} A∩(B∪C)=a* ∩ (a+aa)b* = {a,aa} 5 votes 5 votes Deepak Poonia commented Apr 23, 2022 reply Follow Share $\color{red}{\text{Find Video Solution Below:}}$ Video Solution 3 votes 3 votes parth023 commented May 5, 2023 reply Follow Share for statement $S_{1}$ , let set $A$ be set of odd integers, set $B$ be set of multiple 2 and set $C$ be set of multiple 4. $A\cup (B\cap C$) is empty set. 0 votes 0 votes Please log in or register to add a comment.
–5 votes –5 votes ans should be D. intersection of infinite sets need not be finite. addition of two irrational is also irrational. jayendra answered Dec 31, 2014 jayendra comment Share Follow See 1 comment See all 1 1 comment reply Akash Kanase commented Nov 28, 2015 reply Follow Share Addition of two irrational no can be rational. 2 votes 2 votes Please log in or register to add a comment.