71 votes 71 votes Let $p, q, r, s$ represents the following propositions. $p:x\in\left\{8, 9, 10, 11, 12\right\}$ $q:$ $x$ is a composite number. $r:$ $x$ is a perfect square. $s:$ $x$ is a prime number. The integer $x\geq2$ which satisfies $\neg\left(\left(p\Rightarrow q\right) \wedge \left(\neg r \vee \neg s\right)\right)$ is ____________. Mathematical Logic gatecse-2016-set1 mathematical-logic normal numerical-answers propositional-logic + – Sandeep Singh asked Feb 12, 2016 recategorized Jun 25, 2017 by Silpa Sandeep Singh 12.9k views answer comment Share Follow See all 0 reply Please log in or register to add a comment.
Best answer 130 votes 130 votes $\neg((p → q) \wedge (\neg r \vee \neg s))$ $\quad \equiv (\neg(\neg p \vee q)) \vee (\neg(\neg r \vee \neg s))$ $\quad \equiv (p \wedge \neg q) \vee (r \wedge s)$ which can be read as $(x\in \{8,9,10,11,12\}$ AND $x$ is not a composite number$)$ OR $(x$ is a perfect square AND $x$ is a prime number$)$ Now for $x$ is a perfect square and $x$ is a prime number, this can never be true as every square has at least $3$ factors, $1,x$ and $x^2.$ So, second condition can never be true. which implies the first condition must be true. $x\in \{8,9,10,11,12\}$ AND $x$ is not a composite number But here only 11 is not a composite number. so only $11$ satisfies the above statement. ANSWER $11.$ abhilashpanicker29 answered Feb 14, 2016 edited Jun 7, 2018 by Arjun abhilashpanicker29 comment Share Follow See all 2 Comments See all 2 2 Comments reply manikantsharma commented Jul 17, 2022 reply Follow Share answer 1 thing, satisfy means either tautology or contingency then why only choosing number to prove it as TRUE? 0 votes 0 votes rhl commented Jun 6, 2023 reply Follow Share Actually each of $p, q, r, s$ are predicates with single variable $x$. Since expression given is not quantified so have to replace $x$ with a value to turn the predicate into a proposition. The questions asks for a $x$ which satisfy the given expression. 0 votes 0 votes Please log in or register to add a comment.
8 votes 8 votes ~((p → q) ⋀ (~r ⋁ ~s)) = (~(~p ⋁ q)) ⋁ (~(~r ⋁ ~s)) =(p ⋀ ~q) ⋁ (r ⋀ s) now all whole numbers > 1 are either prime or composite . taking that into consideration.. ~q is equal to s so equation becomes (p ⋀ ~q) ⋁ (r ⋀ s) = (p ⋀ s) ⋁ (r ⋀ s) = ( p ⋁ r ) ⋀ s now to above expression to become tautology .. s should be true... only prime number in our set is 11... So answer would be 11 yogeshwarmisal answered Mar 4, 2017 yogeshwarmisal comment Share Follow See all 0 reply Please log in or register to add a comment.
7 votes 7 votes (p ⇒ q) will give {8, 9, 10, 12} ¬r will give {8, 10, 11, 12} ¬s will give {8, 9, 10, 12} (¬r ∨ ¬s) will give {8, 9, 10, 11, 12} (p ⇒ q) ∧ (¬r ∨ ¬s) will give {8, 9, 10, 12} ¬((p ⇒ q) ∧ (¬r ∨ ¬s)) will give 11. Thus, C is the correct option. Regina Phalange answered May 1, 2017 Regina Phalange comment Share Follow See all 2 Comments See all 2 2 Comments reply arush_verma commented May 20, 2018 reply Follow Share this seems to always work. I'm still hesitant. can you explain why it always works.? 0 votes 0 votes Pranavpurkar commented Dec 31, 2021 reply Follow Share arush_verma we can replace ^ with Ո and ˅ with U. 0 votes 0 votes Please log in or register to add a comment.
1 votes 1 votes Answer 11 abhilashpanicker29 answered Feb 12, 2016 abhilashpanicker29 comment Share Follow See all 2 Comments See all 2 2 Comments reply Debasmita Bhoumik commented Feb 12, 2016 reply Follow Share explain plz 0 votes 0 votes Puja Mishra commented Feb 3, 2017 reply Follow Share First simplify the equation... then take any number from that set and try to get true value of the equation by applying the number... u can see that 11 is satisfying the equation ... 1 votes 1 votes Please log in or register to add a comment.