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Need help in negation of statements

  1. Jul 10, 2011 #1
    Hi

    I have to negate the following statements and then express again
    in English. I need to know if I am making any mistakes.

    a)Everyone who is majoring in math has a friend who needs
    help with his homework.

    b)Everyone has a roommate who dislikes everyone.

    c)There is someone in the freshman class who doesn't
    have a roommate.

    d)Everyone likes someone,but no one likes everyone.

    My answers are as follows--------

    a) Let P(x)= x is majoring in math
    Q(x)=x needs help with homework
    M(x,y)=x and y are friends

    The statement would be

    [tex]\exists x (P(x))\rightarrow \exists y (M(x,y)\wedge Q(y))[/tex]

    So the negated statement would be

    [tex]\neg \left[\exists x (P(x))\rightarrow \exists y (M(x,y)\wedge Q(y))\right][/tex]

    [tex]\neg \left[\neg(\exists x (P(x)) \vee \exists y (M(x,y)\wedge Q(y))\right][/tex]

    [tex]\neg \left[\forall x \neg P(x) \vee \exists y (M(x,y)\wedge Q(y))\right] [/tex]

    [tex]\neg (\forall x \neg P(x)) \wedge \neg \exists y (M(x,y)\wedge Q(y)) [/tex]

    [tex]\exists x P(x) \wedge \forall y \neg (M(x,y)\wedge Q(y))[/tex]

    [tex]\exists x P(x) \wedge \forall y (\neg M(x,y) \vee \neg Q(y))[/tex]

    To translate this statement back to English would be

    Some person is majoring in math and everyone is either not a
    friend of this person or doesn't need help in homework.

    b) let Q(x,y)=x and y are roommates
    M(x,y)=x likes y

    The statement would be

    [tex]\forall x \left[\exists y (Q(x,y) \wedge \forall z(\neg M(y,z)))\right] [/tex]

    so the nagated statement would be

    [tex]\neg \forall x \left[\exists y (Q(x,y)\wedge \forall z(\neg M(y,z)))\right] [/tex]

    [tex]\exists x \forall y \left[\neg Q(x,y) \vee \exists z(M(y,z)) \right] [/tex]

    Translation:

    Either there is some person who is not roommate with anybody or there is
    someone who is liked by all.

    c)let P(x)= x is in freshman class.
    M(x)=x has a roommate.

    The statement would be

    [tex]\exists x \left[ P(x)\wedge \neg M(x) \right] [/tex]

    So the negated statement is

    [tex]\neg \exists x \left[ P(x)\wedge \neg M(x) \right] [/tex]

    [tex]\forall x \left[ \neg P(x) \vee M(x) \right] [/tex]

    Translation:

    Everyone either is not in freshman class or has a roommate.

    d)let M(x,y)= x likes y

    The statement would be

    [tex]\forall x \left[ \exists y (M(x,y))\wedge \exists z \neg M(x,z) \right][/tex]

    The negated statement would be

    [tex]\neg \forall x \left[ \exists y (M(x,y))\wedge \exists z \neg M(x,z) \right][/tex]

    [tex]\neg \left[\forall x \exists y (M(x,y))\wedge \forall x \exists z \neg M(x,z) \right][/tex]

    [tex](\neg \forall x \exists y M(x,y)) \vee (\neg \forall x \exists z \neg M(x,z) )[/tex]

    [tex]\left[\exists x \forall y \neg M(x,y)\right] \vee \left[\exists x \forall z M(x,z)\right][/tex]

    Translation:

    Either there is someone who likes everyone or there is someone who doesn't like
    everyone.



    Please comment

    Thanks
     
  2. jcsd
  3. Jul 10, 2011 #2
    any help ?
     
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