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Logic, Truth Table question

  1. Aug 24, 2015 #1
    << Mentor Note -- Moved from the technical math forums, so no HH Template is shown >>

    I am currently working through a Finite math book Intro to finite math: second Edition Kemeny, Snell, and Thompson. One of the exercises wants me to construct a truth table for the following:
    ~(p|q) earlier I am told that let p|q express that "p and q are not both true". Earlier I worked out that the symbolic form of this statement (p|q) to be ~(p/\~q).

    my work on constructing a truth table for ~(p|q)
    p|q ~ ~ (p /\ ~ q)
    t t F T t f f t
    t f T F t t t f
    f t F T f f f t
    f f F T f f t f

    From here I thought I was to answer the ~ closest to (p, by countering what was under /\, giving me T F T T and then further negating that, ending with F T F F. This is wrong according to the book. as this truth table should end with T F F F.

    2 questions: Is my symbolic form of ~(p|q) wrong and therefore my answer wrong, and/or, did I work something out wrong giving me the wrong answer. I think I probably made the symbolic version wrong but am not sure of how to go about this.
     
    Last edited by a moderator: Aug 24, 2015
  2. jcsd
  3. Aug 25, 2015 #2

    haruspex

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    This doesn't look right to me at all.
    I have always understood p|q to represent "p or q". That isn't "p and q are not both true", which would be (~p)|(~q), or equivalently ~(p^q). It certainly cannot be ~(p/\~q) since that loses the symmetry between p and q.
     
  4. Aug 25, 2015 #3
    I can assure you that this book states :
    7. let p|q express that "p and q are not both true." write a symbolic expression for p|q using ~ and /\. This text book uses the notation \/ for or, /\ for and, ~ for negate.
    I hope this helps and thank you for your response
     
  5. Aug 25, 2015 #4

    haruspex

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    Ok, that's fine, but that does not make it ~(p^~q). As I wrote, that is not symmetric in p and q. Let's get that right first.
     
  6. Aug 26, 2015 #5
    Okay, I do understand that what I did was incorrect. I am just trying to figure out how to solve this. I am having trouble tying to break it down in to simple statements (p|q) = p and q are not both true. Simple statements: p is true, q is true. Here i don't know what to do about the both clause, because i don't think these two statements represent the given symbolic statement. I tried using exclusive disjunction ~ (p|q) = ~(p \/ q) but this truth table did not math what the answer in the book says. the book says the truth table should be TFFF.

    Any insight on what I am doing wrong, or missing?
     
  7. Aug 26, 2015 #6

    haruspex

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    Right. If they are not both true then at least one of them is. ...?
    Can you put that into English in the form "either .... or ...."?
     
  8. Aug 27, 2015 #7

    andrewkirk

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    I cannot understand the truth table in the OP because the column headings are all over the place.
    However "p and q are not both true" means the same as "not (p and q are both true)", which is symbolised as ##\neg(p\wedge q)##. So you can make a truth table by just reversing every entry in the standard truth table for the conjunction ##p\wedge q##.

    As I understand it, you have been asked to symbolise the negation of p|q, in other words ##\neg(\neg(p\wedge q))##. If you are using Classical Logic, rather than some fancy version like Intuitionist or Minimal Logic, you can use 'double negation elimination' to simlify that.
     
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