53 votes 53 votes What is the logical translation of the following statement? "None of my friends are perfect." $∃x(F (x)∧ ¬P(x))$ $∃ x(¬ F (x)∧ P(x))$ $ ∃x(¬F (x)∧¬P(x))$ $ ¬∃ x(F (x)∧ P(x))$ Mathematical Logic gatecse-2013 mathematical-logic easy first-order-logic + – Arjun asked Sep 24, 2014 • edited Jun 8, 2018 by kenzou Arjun 14.3k views answer comment Share Follow See all 3 Comments See all 3 3 Comments reply Mk Utkarsh commented Feb 15, 2018 reply Follow Share A) "Some of my friends are not perfect" B) "Some are not my friends and are perfect" C) "Some are not my friends and are not perfect" D) "There exist no one who is my friend and is perfect" 12 votes 12 votes IamDRD commented Jan 24, 2021 reply Follow Share can we solve it using truth table? 0 votes 0 votes ankit3009 commented Apr 24, 2021 reply Follow Share Yes but it'll take lot of time 0 votes 0 votes Please log in or register to add a comment.
Best answer 65 votes 65 votes some of my friends are not perfect some of those who are not my friends are perfect some of those who are not my friends are not perfect NOT (some of my friends are perfect) / none of my friends are perfect Correct Answer: $D$ Bhagirathi answered Sep 26, 2014 • edited Apr 24, 2019 by Naveen Kumar 3 Bhagirathi comment Share Follow See all 7 Comments See all 7 7 Comments reply Arjun commented Sep 26, 2014 i edited by Puja Mishra Jan 3, 2018 reply Follow Share second part of (D) is correct. But "NOT all of my friends are perfect" is not correct. It means some of my friends are not perfect but actually none of my friends are perfect. So the correct way to put (D) would be "There does not exist any friend of mine who is perfect" or "none of my friends are perfect" . For (B) and (C) you should replace "those" with "some of those" because by default it will take "All" as per English grammar. 15 votes 15 votes Bhagirathi commented Sep 26, 2014 reply Follow Share edited now........ yah you are right i must have written some of those 3 votes 3 votes Arjun commented Sep 26, 2014 reply Follow Share ok :) NOT (all of my actually it should be NOT (some of my 6 votes 6 votes Bhagirathi commented Sep 26, 2014 reply Follow Share got it atlast 0 votes 0 votes Vinil commented Nov 7, 2017 i edited by JAINchiNMay Oct 28, 2022 reply Follow Share @ arjun sir, Can we retransform the given statement like this...."None of my friends are perfect" to "All my freinds are not perfect" so that we can directly write like this, for ∀x( F(X)-->~P(X)) .... What do you say? Please help 2 votes 2 votes Swapnil Naik commented Jan 15, 2019 reply Follow Share @Arjun @Bhagirathi None of my friends are perfect can be written an "all of my friends are not perfect". Now all of my friends are not perfect could be written as: $\forall x (F(x) \implies \lnot P(x))$ $\forall x (\lnot F(x) \cup \lnot P(x))$ By de Morgan's law: - $\lnot \exists(F(x) \cap P(x))$ Which is equivalent to option D. Sir, what is the right way to interpret it? 5 votes 5 votes ghostman23111 commented Aug 25, 2019 reply Follow Share We can also do this: "Atleast one of my is perfect" $\exists x(F(x) \wedge P(x) )$ and Negation of this will be $\neg \exists x(F(x) \wedge P(x)) = \forall x\neg(F(x)\wedge P(x))= \forall x (\neg F(x) \lor \neg P(x))$ 0 votes 0 votes Please log in or register to add a comment.
48 votes 48 votes $F\left(x\right): x \text{ is my friend.}$ $P\left(x\right):\text{x is perfect.}$ $\text{“None of my friends are perfect"}$ can be written like $\forall x[F(x)\implies\neg P(x)]$ $\equiv \forall x[\neg F(x)\vee \neg P(x)]$ $\equiv \forall x\neg[F(x)\wedge p(x)]$ $\equiv \neg\exists x[F(x)\wedge P(x)]$ So, the answer is D. vnc answered Nov 10, 2015 • edited Apr 2, 2018 by Soumya29 vnc comment Share Follow See all 7 Comments See all 7 7 Comments reply Show 4 previous comments sameer1992 commented Aug 23, 2017 reply Follow Share @vnc Can u elaborate after 2nd last step? 0 votes 0 votes Sunidhi chauhan commented Dec 10, 2017 reply Follow Share why you again negate theirexist?????in last line??? plz tell me my approach is right or not for all people in the world who is my friend then not perfect i.e ∀x[F(x)--->~P(x)] =∀x[~F(x)∨~P(x)] =∀x~[F(x)∧p(x)] now according to me last line should be... =∃x[F(x)∧P(x)] but why you put negation?????? 0 votes 0 votes Puja Mishra commented Jan 3, 2018 reply Follow Share Wrong calculation ... correct it ... 1 votes 1 votes Please log in or register to add a comment.
10 votes 10 votes "None of my friends are perfect." Lakshman Bhaiya answered Feb 19, 2018 Lakshman Bhaiya comment Share Follow See all 0 reply Please log in or register to add a comment.
9 votes 9 votes "None of my friends are perfect." It is NOT the complement of "All of my friends are perfect" So A is not the answer. (A frequently done mistake) It is the complement of "At least one of my friend is perfect" So D is the answer. Ahwan answered Jan 18, 2018 • edited Jan 18, 2018 by Ahwan Ahwan comment Share Follow See all 2 Comments See all 2 2 Comments reply Abhishek Gupta 1 commented Oct 2, 2018 reply Follow Share If I take complement of "All of my friends are perfect" then what will it mean? What will be the semantic meaning of it. 0 votes 0 votes Ahwan commented Oct 2, 2018 reply Follow Share @ Abhishek Gupta 1 "At least one of my friends is not perfect." 1 votes 1 votes Please log in or register to add a comment.