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Translating a statement to logic.

  1. Jan 28, 2012 #1
    1. The problem statement, all variables and given/known data

    Translate the following sentences into propositional or predicate logic. Use the shorthand
    symbols (e.g. [itex]\vee[/itex]) and define the meaning of each of your predicates and propositional variables. Be sure to include a domain (aka replacement set) for each quantified variable. You may need to rewrite the sentences slightly so as to make a variable more explicit.

    "Irrational numbers have decimal expansions that neither terminate nor become
    periodic".

    2. Relevant equations



    3. The attempt at a solution

    Since the predicate refers to the decimal expansion of irrational numbers, I don't know whether to (a) declare/quantify a variable that represents the decimal expansion of an irrational number, or to (b) declare/quantify a variable that represents an irrational number.

    (a)
    T(x) is "d terminates".
    P(x) is "d becomes periodic".
    D is the set of all decimal expansions of irrational numbers.

    [itex]\left(\forall d \in D\right)\left(\neg T(d) \wedge \neg P(d)\right)[/itex]

    ...versus:
    (b)
    D(x) is “x has a decimal expansion”.
    T(D(x)) is “the decimal expansion of x terminates”.
    P(D(x)) is “the decimal expansion of x becomes periodic”.
    I is the set of all irrational numbers.

    [itex]\left(\forall x \in I\right)\left( D(x) \wedge \neg T(D(x)) \wedge \neg P(D(x)) \right)[/itex]
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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