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Math Help - Equivalence of terms

  1. #1
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    Equivalence of terms

    We have

    A = ∀xP(x) ⊻ ∀yQ(y)
    B = ∃xP(x) ⇔ ∀yQ(y)

    Are A and B equal? I get that they are.

    Is it correct? I have solved it like this:

    First we negate the B:

    ∃xP(x) ⇔ ∀yQ(y)
    ∃xP(x) ⇔ ∀yQ(y)

    then we get

    ∀xP(x) ⇔ ∃yQ(y)

    And then we insert something. (for example):

    P(x) x is divisible by 2
    Q(y) y is not divisible by 2

    For both A and not B i get zero. So they are equal.

    Is my thinking correct?
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  2. #2
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    First, A and ~B are not equal, since they are not the same formulas. A formula is a string of symbols and formulas are equal if and only if they are the same string of symbols. However, they might be logically EQUIVALENT.

    So the question is whether A and ~B are logically equivalent. (Should I surmise that the connective in your A is a typo and is supposed to be the "if and only if" sign just as is the connective in B?)

    Second, you did not properly negate B. Indeed, what you thought was negating B is just an equivalent (so very much not a negation) of B.

    Third, you don't show equivalence by using a particular interpretation of the predicates, such as you interpreted P and Q. You can use particular interpretations to prove the invalidy of a formula (here the formula in question is A <-> ~B), but not of the validty of a formula, since validity depends on the formula being true with ANY interpretation of the predicates.

    Those three points suggest to me that you need a very thorough review of the fundamentals of symbolic logic.
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  3. #3
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    OK, thank you for this introduction.

    How would I properly negate B?

    So now I know what do I have to proof:

    ∀xP(x) ⊻ ∀yQ(y) ⇔ (∃xP(x) ⇔ ∀yQ(y))

    Sorry if I don't know, but I don't have any book for predicates. Do you know a website with examples and some introdoction for predicate.
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  4. #4
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    The negation of any formula P is ~P, so since B is ExPx <-> AyQy, the negation of B is ~(ExPx <-> AyQy).

    What does the symbol



    stand for?

    /

    For a good book, get "Logic: Techniques Of Formal Reasoning" by Kalish, Montague, and Mar?

    Are you taking a course that has no textbook specified?
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  5. #5
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    Quote Originally Posted by MoeBlee View Post
    What does the symbol
    ⊻ stand for?
    Usually \underline  \vee  is use for the exclusive or.
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  6. #6
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    Ah, I hadn't see that one before. Thanks.
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