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Math Help - Propositional Logic

  1. #1
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    Propositional Logic

    George, new coach of the All Stars team, reveals the following news about team selection:

    “When Daniel hasn’t had breakfast, Fred plays in midfield”
    “Greg is not in goal only if William is on the substitutes’ bench”

    “Fred playing in midfield means that Mark switches to left flank”
    “If Daniel has had breakfast, William is omitted from the substitutes’ bench”
    “Given Harry is injured, then Mark cannot switch to left flank”

    (a) Formulate these five facts as statements in propositional logic.

    (b) Is it true that either Fred plays in midfield or Greg is in goal?

    (c) Can one deduce that William makes the substitutes’ bench or Mark cannot switch to left flank?

    (d) Does Greg not being in goal imply that the Harry is fit?



    ---------------------------------------------------------------------------------


    So I'm trying to do these questions and I can't entirly get my head around it:


    for a) I have:

    I'll use - as the negation.
    -d > f
    -g > w
    f > m
    d > -w
    n > -m


    and for b) i have tried linking together statements and expanding using the implication laws eg:
    (dVf)^(gVw)
    but don't know where to go from there.
    Can someone help me as to how to go about solving these problems?
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  2. #2
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    Quote Originally Posted by AdamC View Post
    -d > f
    -g > w
    f > m
    d > -w
    n > -m
    By n do you denote "Harry is injured"? Probably, h would be better. In general, it is also good for propositional letter to write which statement it denotes. For example, it may not be immediately clear whether h means "Harry is injured" or "Harry is fit". Otherwise, I agree with your formulation.

    Concerning (b) -- (d), it is not clear whether you need to use truth tables or syntactic laws and, if the latter, then which laws. First, I would write the cotrapositives of each given formula with double negations removed. Below I'll list all given formulas with their contrapositives.

    (1) -d -> f; -f -> d
    (2) -g -> w; -w -> g
    (3) f -> m; -m -> -f
    (4) d -> -w; w -> -d
    (5) h -> -m; m -> -h

    Then I would prove -f -> g, which is equivalent to f \/ g. From -f we get d by (1), -w by (4) and g by (2).

    The answer to (d) is yes. I went the opposite way: to derive -h it is sufficient to show m by (5), which follows from f by (3) and so on until I arrived at -g.
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