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Math Help - Want to negate the following statements

  1. #1
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    Want to negate the following statements

    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

    \exists x (P(x))\rightarrow \exists y (M(x,y)\wedge Q(y))

    So the negated statement would be

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

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

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

    \neg (\forall x \neg P(x)) \wedge \neg \exists y (M(x,y)\wedge Q(y))

    \exists x P(x) \wedge \forall y \neg (M(x,y)\wedge Q(y))

    \exists x P(x) \wedge \forall y (\neg M(x,y) \vee \neg Q(y))

    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

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

    so the nagated statement would be

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

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

    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

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

    So the negated statement is

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

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

    Translation:

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

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

    The statement would be

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

    The negated statement would be

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

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

    (\neg \forall x \exists y M(x,y)) \vee (\neg \forall x \exists z \neg M(x,z) )

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

    Translation:

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



    Please comment

    Thanks
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  2. #2
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    Re: Want to negate the following statements

    Quote Originally Posted by issacnewton View Post
    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.

    Please comment
    You are making too much out of this for my tastes, but maybe that is required.

    Examples:
    a) Someone is majoring in mathematics and none of his friends needs help with homework.

    c) Everyone in the freshman class has a roommate.
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  3. #3
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    Re: Want to negate the following statements

    Thanks Plato for reply.

    Where am I making mistakes then ? Thats what I want to know. I am using standard procedure as given in Daniel Velleman's book
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  4. #4
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    Re: Want to negate the following statements

    Quote Originally Posted by issacnewton View Post
    Where am I making mistakes then ? Thats what I want to know. I am using standard procedure as given in Daniel Velleman's book
    As I did say, you may be required to do all of that symbolic workings. Frankly, you loose me in your process. I don't understand why for instance you introduce a non-friend in part (a)?
    You have to admit that my negation is a natural way of saying that.
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  5. #5
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    Re: Want to negate the following statements

    So Plato, in part a), are my symbolic equations correct ? Or is it just the problem with the translation back in English ? In part a) , which equation do you
    think I took wrong step ?
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  6. #6
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    Re: Want to negate the following statements

    Quote Originally Posted by issacnewton View Post
    So Plato, in part a), are my symbolic equations correct ?
    I your OP you said:
    Quote Originally Posted by issacnewton View Post
    I have to negate the following statements and then express again in English. I need to know if I am making any mistakes.
    There is nothing in the directions about putting these into symbolic form. So I ask you if you are required to do so? If not why do it?
    It seems to me the directions “to negate the following statements and then express again in English” are very straightforward. So once again, I do not follow your symbolic translations. So about the symbols I cannot say. Sorry.
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  7. #7
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    Re: Want to negate the following statements

    Oh I am sorry Plato. I probably didn't say it correctly. Yes , I am required to put this in the form of symbolic equations, complete the negation using
    various rules about the quantifiers and then translate back into the English. The book "How to Prove it" is teaching in chapter 2 , the use of
    quantifiers. Author did give some examples and I am doing the exercises....
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  8. #8
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    Re: Want to negate the following statements

    Quote Originally Posted by issacnewton View Post
    I am required to put this in the form of symbolic equations, complete the negation using various rules about the quantifiers and then translate back into the English.
    Quote Originally Posted by issacnewton View Post
    a) Let P(x)= x is majoring in math
    Q(x)=x needs help with homework
    M(x,y)=x and y are friends
    Well the symbolic forms complicate things.
    Using your notation the a) statement becomes:
    \left( {\forall x} \right)\left[ {P(x) \to \left( {\exists y} \right)\left[ {M(x,y) \wedge Q(y)} \right]} \right]

    The negation of which is \left( {\exists x} \right)\left[ {P(x) \wedge \left( {\forall y} \right)\left[ {\neg M(x,y) \vee \neg Q(y)} \right]} \right].

    As you no doubt can see translating that back to ‘natural’ English is complicated.
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  9. #9
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    Re: Want to negate the following statements

    Yes Plato, I see the point. I think author just wants the students to get the hang of it I guess. But the way you have written the affirmative part of part a) looks different from the way I have done. Or I am doing the same thing ?

    Probably the mathematical statements , when negated , won't be so hard to interpret to us mortals......
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    Re: Want to negate the following statements

    Quote Originally Posted by issacnewton View Post
    But the way you have written the affirmative part of part a) looks different from the way I have done. Or I am doing the same thing ?
    Of course it is different. It is correct. I told you that I could not follow what you did much less why you did it.
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  11. #11
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    Re: Want to negate the following statements

    Thanks. So I am correct but did differently. These quantifiers are really brain twisters I guess. But understanding it is of great importance to appreciate
    the logical structure in mathematics. One of the books which I got is Copis's Introduction to Logic. Apart from symbolic logic, he goes in great detail about
    discussing general logic , which is applied in various fields. So is mathematical logic a subset of philosophical/general logic ? Because, Copi discusses lot of
    things ( usually having some Latin/Greek names) without any symbols.
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    Re: Want to negate the following statements

    Quote Originally Posted by issacnewton View Post
    So I am correct but did differently.
    No. But now I give up.
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  13. #13
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    Re: Want to negate the following statements

    Quote Originally Posted by Plato View Post
    Of course it is different. It is correct.
    I thought that you meant I was correct. some misunderstanding ?
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  14. #14
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    Re: Want to negate the following statements

    First, I pointed out in another thread that it's easier to check your work if you choose mnemonic names for predicates.

    Quote Originally Posted by issacnewton View Post
    a)Everyone who is majoring in math has a friend who needs
    help with his homework
    The statement would be

    \exists x (P(x))\rightarrow \exists y (M(x,y)\wedge Q(y))
    The closing parenthesis after P(x) should be moved to the end of the formula.

    \exists x P(x) \wedge \forall y (\neg M(x,y) \vee \neg Q(y))

    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.
    Correct. I prefer to turn the disjunction \neg M(x,y) \vee \neg Q(y) into an implication M(x,y)\to\neg Q(y), so the English version is "Someone is majoring in mathematics and none of his friends needs help with homework," as Plato said. It is more natural to say "All people are mortal" (using an implication) than "Everyone is either not a person or is mortal" (using a disjunction).

    b)Everyone has a roommate who dislikes everyone.

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

    The statement would be

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

    so the nagated statement would be

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

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

    Translation:

    Either there is some person who is not roommate with anybody or there is someone who is liked by all.
    The negated formula is correct; the English translation is not. The formula corresponding to the English phrase is

    (\exists x \forall y \neg Q(x,y)) \vee (\exists x \forall y\,M(y,x))

    Again, I prefer to turn \neg Q(x,y) \vee \exists z(M(y,z)) into Q(x,y) \to \exists z(M(y,z)). Then the English translation is "There is a person such that all of his/her roommates love somebody."

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

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

    The statement would be

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

    So the negated statement is

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

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

    Translation:

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

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

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

    The statement would be

    \forall x \left[ \exists y (M(x,y))\wedge \exists z \neg M(x,z) \right]
    I would say the direct formalization is

    (\forall x \exists y M(x,y))\wedge (\forall x\neg\forall z M(x,z))

    This is equivalent to your formula, but it is closer to English because the main connective is a conjunction, which corresponds to "but."

    The negated statement would be

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

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

    (\neg \forall x \exists y M(x,y)) \vee (\neg \forall x \exists z \neg M(x,z) )

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

    Translation:

    Either there is someone who likes everyone or there is someone who doesn't like
    everyone.
    The negated formula is correct. Concerning English, I think "there is someone who doesn't like everyone" means \exists x \neg\forall y M(x,y). To express \exists x \forall y \neg M(x,y), I would say, "there is someone who doesn't like anybody."
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  15. #15
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    Re: Want to negate the following statements

    Priviet Makarov

    Thanks for the detailed response. Which introductory book on logic did you use as an undergrad ?
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