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Math Help - Predicate Logic converted to English Statements

  1. #1
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    Predicate Logic converted to English Statements

    Could someone please help me with these predicate logic statements:

    domain: all students in your class
    C(x): "x has a cat"
    D(x): "x has a dog"
    F(x): "x has a ferret"

    Some student in your class has a cat and a ferret, but not a dog.
    <br />
\exists{x[C(x)\wedge}{F(x)\wedge}{\neg}{D(x)]}<br />

    No student in your class has a cat, a dog, and a ferret.
    <br />
\forall{\neg}{[C(x)\wedge}{D(x)\wedge}{F(x)]}<br />

    For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has one of these animals as pets.
    <br />
\forall{x}\exists{x[C(x)\vee}{D(x)\vee}{F(x)]}<br />
    A little unsure about this one.


    The next question:

    P(x): "x is a duck"
    Q(x): "x is one of my poultry"
    R(x): "x is an officer"
    S(x): "x is willing to waltz"

    No ducks are willing to waltz.
    <br />
\forall{x[P(x)\rightarrow}{\neg}{S(x)]}<br />

    No officers ever decline to waltz.
    <br />
\forall{x[}{\neg}{R(x)\rightarrow}{\neg}{S(x)]}<br />

    All my poultry are ducks.
    <br />
\forall{x[Q(x)\rightarrow}{P(x)]}<br />

    My poultry are not officers.
    <br />
\forall{x[Q(x)\rightarrow}{\neg}{R(x)]}<br />
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  2. #2
    MHF Contributor
    Grandad's Avatar
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    Predicate Logic

    Hello aaronrj
    Quote Originally Posted by aaronrj View Post
    Could someone please help me with these predicate logic statements:

    domain: all students in your class
    C(x): "x has a cat"
    D(x): "x has a dog"
    F(x): "x has a ferret"

    Some student in your class has a cat and a ferret, but not a dog.
    <br />
\exists{x[C(x)\wedge}{F(x)\wedge}{\neg}{D(x)]}<br />
    Correct.

    No student in your class has a cat, a dog, and a ferret.
    <br />
\forall\color{red}x\color{black}{\neg}{[C(x)\wedge}{D(x)\wedge}{F(x)]}<br />
    Correct, except that you missed out the x.

    For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has one of these animals as pets.
    <br />
\forall{x}\exists{x[C(x)\vee}{D(x)\vee}{F(x)]}<br />
    A little unsure about this one.
    No. You can re-phrase it as "Someone owns a cat AND someone owns a dog AND someone owns a ferret". So you get:

    \exists x[C(x)] \land \exists x ...

    I'm sure you can complete it now.



    The next question:

    P(x): "x is a duck"
    Q(x): "x is one of my poultry"
    R(x): "x is an officer"
    S(x): "x is willing to waltz"

    No ducks are willing to waltz.
    <br />
\forall{x[P(x)\rightarrow}{\neg}{S(x)]}<br />
    Correct.

    No officers ever decline to waltz.
    <br />
\forall{x[}{\neg}{R(x)\rightarrow}{\neg}{S(x)]}<br />
    No. This means "If you're not an officer, then you're not willing to waltz."

    "No officers ever decline to waltz" simply means "Every officer is willing to waltz". So

    \forall x [R(x)\rightarrow S(x)]

    All my poultry are ducks.
    <br />
\forall{x[Q(x)\rightarrow}{P(x)]}<br />

    My poultry are not officers.
    <br />
\forall{x[Q(x)\rightarrow}{\neg}{R(x)]}<br />
    Both of these are correct!

    Grandad
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