51 votes 51 votes Which one of the following propositional logic formulas is TRUE when exactly two of $p,q$ and $r$ are TRUE? $(( p \leftrightarrow q) \wedge r) \vee (p \wedge q \wedge \sim r)$ $( \sim (p \leftrightarrow q) \wedge r)\vee (p \wedge q \wedge \sim r)$ $( (p \to q) \wedge r) \vee (p \wedge q \wedge \sim r)$ $(\sim (p \leftrightarrow q) \wedge r) \wedge (p \wedge q \wedge \sim r) $ Mathematical Logic gatecse-2014-set1 mathematical-logic normal propositional-logic + – go_editor asked Sep 28, 2014 • edited Jun 20, 2017 by Silpa go_editor 13.8k views answer comment Share Follow See all 5 Comments See all 5 5 Comments reply Show 2 previous comments Kiyoshi commented Dec 20, 2021 reply Follow Share Combination which gives False on exactly two of p,q,r are true. Option A --- (P,Q,R) = (F,T,T) Option C--- (P,Q,R) = (T,F,T) Option D--- (P,Q,R) = (T,T,F) or (T,F,T) or (F,T,T) Now, option B these combination give true.. Option B --- (P,Q,R) = (T,T,F) or (T,F,T) or (F,T,T) 2 votes 2 votes PSYBIL commented Oct 18, 2023 reply Follow Share Ambiguous Question 0 votes 0 votes ritiksri8 commented 3 days ago reply Follow Share for all the cases values needs to be true. case 1 : P=T Q=T R=F case 2 : P=T Q=F R=T case 3: P=F Q=T R=T 0 votes 0 votes Please log in or register to add a comment.
3 votes 3 votes Correct answer is option B. Mostafize Mondal answered Dec 3, 2018 Mostafize Mondal comment Share Follow See all 0 reply Please log in or register to add a comment.
2 votes 2 votes Consider 4,6,7 cases and check options. $B)\\ (\sim (p\leftrightarrow q)\wedge r) \vee (p \wedge q \wedge \sim r)\\$ $4^{th}$ case $=(\sim (F\leftrightarrow T)\wedge T) \vee (F \wedge T \wedge \sim T)\\ =(\sim F\wedge T) \vee F\\ =(T\wedge T) \vee F\\ =T \vee F\\ =T$ Similarly Check $6^{th} \&\ 7^{th}$ cases. Correct Answer: B KUSHAGRA गुप्ता answered Jul 2, 2020 KUSHAGRA गुप्ता comment Share Follow See all 0 reply Please log in or register to add a comment.
0 votes 0 votes In option (B) : We assume ‘p’ and ‘q’ to be TRUE. Therefore, ‘r’ is false because exactly two of ‘p’, ‘q’, and ‘r’ can be TRUE. p <--> q means if both ‘p’ and ‘q’ have same values then p <--> q is TRUE. So, ((p <--> r ) and r) evaluates to be FALSE. Therefore, ~((p <--> r ) and r) is TRUE. Here, two sub- expression are connected via OR operator and one of the sub-expression is TRUE. So, the complete expression becomes TRUE. Therefore, option (B) evaluates to be TRUE. Regina Phalange answered Apr 7, 2017 Regina Phalange comment Share Follow See all 0 reply Please log in or register to add a comment.
0 votes 0 votes If you check for 110 A,B, C all are true. If you check for 111, A, C are true, while B is false. So, B is the answer. Himanshu Kumar 1 answered Nov 3, 2018 Himanshu Kumar 1 comment Share Follow See all 0 reply Please log in or register to add a comment.