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Math Help - one-point rule using Z

  1. #1
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    one-point rule using Z

    Hello guys!

    First of all i would like to express my happiness of the forum coming back live. Nothing is better than this .

    I'm not quite sure if anyone here has used Z to represent discrete mathematics here before. But this is the notation which i've been taught at the uni. But since i'm not having much of a help on this. I'm a bit feeling a bit left alone. I got to finish this assignment off I about week time.

    Could anyone please help me point out some direction on where to start? Which inference rule to apply first to simplify the following quantification using one-point rule and predicate logic

    \exists x:\mathbb{N}\bullet(\forall y:\mathbb{N}\bullet y\neq z \vee y \neq x) \Rightarrow (\forall y:\mathbb{N} \bullet z > y)

    'z' is assumed to be an integer.

    As you can see in the predicate part the implication holds. But i'm quite struggling to understand what y \neq z) is actually represnting. 'z' can be an positive or a negative integer. If y is not equal to z meaning that y is always above 0 as its of type natual number.

    thanks a lot guys

    ssharish
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  2. #2
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    Re: one-point rule using Z

    What does "one-point rule using Z" mean?
    In both of your postings, the logical symbols appear to be out of main stream logical notation.
    Perhaps you should post a guide to their meaning.
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  3. #3
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    Re: one-point rule using Z

    Thanks for the reply Plato, I completely agree on why this notation seems to be quite unclearly. I rather have been in an understanding that the notation which Iím using is rather nonstandard. But quite not sure as to why my professor seem to use rather a nonstandard notation to represent the quantification and other discrete logic equation. We study formal methods in which we use Z notation.

    The link below is the book which Iím referring to and also why professor uses it for his teaching. I donít expect you to go through all the whole book. But thatís there for your reference.
    http://www.cs.cmu.edu/~15819/zedbook.pdf

    Out of curiosity, what is a standard notation. Is there a material which i can go through. I'm sorry this might be a simple question. Forgive my silliness.

    One point rule is one of the inference rule which actually states as follow

    If
    the identity of a bound variable is revealed within the quanti
    ed expression,
    then we may replace all instances of that variable, and remove the existential
    quanti
    er. Consider the following predicate:

    \exits x : a \bullet p \wedge x = t

    This states that there is a value of x in a for which p ^x Ét is true. If t is in a,
    and p holds with t substituted for x, then t is a good candidate for this value.

    Refer page 48 from the above link.

    thanks

    ssharish
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    Re: one-point rule using Z

    Quote Originally Posted by ssharish View Post
    The link below is the book which Iím referring to and also why professor uses it for his teaching. I donít expect you to go through all the whole book. But thatís there for your reference.
    http://www.cs.cmu.edu/~15819/zedbook.pdf
    I see from the link that that is Computer Science at Carnegie Mellon.
    That explains a lot. Perhaps you can find a board such as this devoted to computer science.

    As for standard notation, look to books by Irving Copi (a Russel student) or books by Willard Quine.
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  5. #5
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    Re: one-point rule using Z

    this appears to be a logic problem, though, not a computer science problem. i could be wrong (i haven't looked at the referenced text in detail) but here : (the colon) is used instead of "is an element of", and the dot just means "such that" or "it is the case that".
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  6. #6
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    Re: one-point rule using Z

    Hi all, yes Deveno this is definitely a logic question. But i cant get my head around as to what the quantifier suppose to mean. If i try interpret that into English, could anyone please validate if at all it makes any sense or its just me who not thinking it right. So basically it says

    "there exists a x in Natural numbers \exists x : \mathbb{N} such that \bullet for all y in natural number such that y is not equal to z or y is not equal to x \forall y : \mathbb{N} \bullet y \neq z \vee y \neq x implies \Rightarrow for all y in natural numbers such that z is greater than y \forall y : \mathbb{N} \bullet z > y

    z is assumed to be an integer \mathbb{Z}"

    What I donít understand is the predicate y not equal to z or y not equal to x. This just doesnít make sense. z here can have value of both positive and negative in range. Does this mean that y is proving that it will always be positive? If that was the case, we have already proved that because y is of type natural number which means that it canít hold a value bellow 0!

    thanks

    ssharish
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    Re: one-point rule using Z

    Quote Originally Posted by ssharish View Post
    Could anyone please help me point out some direction on where to start? Which inference rule to apply first to simplify the following quantification using one-point rule and predicate logic

    \exists x:\mathbb{N}\bullet(\forall y:\mathbb{N}\bullet y\neq z \vee y \neq x) \Rightarrow (\forall y:\mathbb{N} \bullet z > y)
    What exactly do you want to do with this expression: prove it? It is true when z is a natural number because then for x = z both the premise \forall y:\mathbb{N}.\, y\neq z \vee y \neq x and the conclusion \forall y:\mathbb{N}.\, z > y are false. However, when z is negative, it is false because the conclusion is still false, but the premise is true: every natural number y is different from z.

    Quote Originally Posted by ssharish View Post
    What I donít understand is the predicate y not equal to z or y not equal to x. This just doesnít make sense. z here can have value of both positive and negative in range. Does this mean that y is proving that it will always be positive?
    I am not sure if Z considers y = z a well-formed formula when y\in\mathbb{N} and z\in\mathbb{Z}. But if it does, then I think its meaning is obvious: y and z have the same numerical value.
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