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Math Help - Propositional Logic and Predicates

  1. #1
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    Propositional Logic and Predicates

    Math Predicates and quantifiers?

    Let P(x, y) be the statement "x loves y", where the domain for both x and y is the set of all people in the world. Use quantifiers ( ∀, ∃) to express each of the following statements:

    (a) Everybody loves Raymond.
    (b) Everybody loves somebody.
    (c) There is somebody whom everybody loves.
    (d) There is somebody whom no one loves.
    (e) Nobody loves everybody.
    (f) Everybody loves himself.
    (g) Everybody loves everybody.
    (h) Someone loves at least two people.
    (i) Someone loves exactly two people.

    Thanks for the help.
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  2. #2
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    You really should that you have put some effort into doing what are after all your problems.

    I will do one of the more difficult ones for you.
    (e) \left( {\forall x} \right)\left( {\exists y} \right)\left[ {\neg L(x,y)} \right].

    Now you try the others. Post the results so we can help on them.
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  3. #3
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    Let P(x, y) be the statement x loves y", where the domain for both x and y is the set of
    all people in the world. Use

    (a) Everybody loves Raymond.

    xP(x, Raymond)

    (b) Everybody loves somebody.

    xy(x, y)


    (c) There is somebody whom everybody loves.

    xyP(x, y)


    (d) There is somebody whom no one loves.

    xyP(y, x)


    (e) Nobody loves everybody.

    xyP(x, y)


    (f) Everybody loves himself.

    xP(x, x)


    (g) Everybody loves everybody.

    xyP(x, y)


    (h) Someone loves at least two people.

    ??

    (i) Someone loves exactly two people.

    ??
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  4. #4
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    Part c should be \left( {\exists x} \right)\left( {\forall y} \right)\left[ {P(y,x)}\right].

    Part g should be \left( {\forall x} \right)\left( {\forall y} \right)\left[ {P(x,y)} \right].

    h) \left( {\exists x} \right)\left( {\exists y} \right)\left( {\exists z} \right)\left[ {P(x,y) \wedge P(x,z) \wedge y \ne z} \right].

    i) \left( {\exists x} \right)\left( {\exists y} \right)\left( {\exists z} \right)\left( {\forall w} \right)\left[ {P(x,y) \wedge P(x,z) \wedge y \ne z \wedge \left[ {w \ne y \wedge  \ne z \Rightarrow \neg P(x,w)} \right]} \right].

    The others look correct.
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