Frenchie Posted January 23, 2012 Share Posted January 23, 2012 (edited) Hello all, I'm seeking help in logical equivalence for the following: {[(p^q)V(p^r)]V(q^r)}^~[(p^q)^r] It's asking to have an equivalence of that equation using only negations of conditionals(biconditionals too I assume because of the v) First I take (p∧q), (p∧r), and (q∧r) and (p∧q) which are all conjunctions and change them into the negation of a biconditional based on ^ being the same as <-> which is a conditional so that they are equivalent. ~(p->~q), ~(p->~r), ~(q->~r), ~(p->~q) which gives me the new equation of {[~(p->~q)v~(p->~r)v~(q->~r)]}∧~[~(p->~q)∧r] I then take the remaining ∧ which are {[~(p->~q)v~(p->~r)v~(q->~r)]}∧~[~(p->~q)∧r] and ~[~(p->~q)∧r and do the same as previously. I then take the disjunction two at a time starting with ~(p->~q)v~(p->~r) and change it into the negation of a conditional based on v being the same as -> which is a biconditional so that they are equivalent. I am not sure how to do this step. Afterwards I would take the result of the negation of my conditional which is a biconditional as done above and do it's negation of conditional with the remaining ~(q->~r). I am not sure how to do this step. I appreciate all the help that i can receive, thank you. Thank you for any help! Edited January 24, 2012 by Frenchie Link to comment Share on other sites More sharing options...
Frenchie Posted January 24, 2012 Author Share Posted January 24, 2012 I really need help on this one if anyone can that would be great : / Link to comment Share on other sites More sharing options...
DrRocket Posted January 24, 2012 Share Posted January 24, 2012 Hello all, I'm seeking help in logical equivalence for the following: {[(p^q)V(p^r)]V(q^r)}^~[(p^q)^r] It's asking to have an equivalence of that equation using only negations of conditionals(biconditionals too I assume because of the v) Maybe if you draw a Venn diagram you can see an answer. You are looking for everything that is in two of p,q, and r but not in all three. Link to comment Share on other sites More sharing options...
khaled Posted January 29, 2012 Share Posted January 29, 2012 (edited) This what I get when simplifying your statement: = { [ (p^q) V (p^r) ] V (q^r) } ^ ~[(p^q) ^ r] = { (p^q) V (p^r) V (q^r) } ^ { ~p V ~q V ~r } = { (p^q) V (p^r) V (q^r) } ^ ~{ p ^ q ^ r } So, it's just as DrRocket mentioned, Check: Wolfram|Alpha:Solution Edited January 29, 2012 by khaled Link to comment Share on other sites More sharing options...
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