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Math Help - Is N dense in R?

  1. #1
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    Is N dense in R?

    Is N dense in R?
    Let say (a,b)=(0,1) How come it does not contain an element of ℕ ?
    Is it because it's an open interval?
    Can someone please explain this to me? Also when would it be dense?
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  2. #2
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    Re: Is N dense in R?

    Quote Originally Posted by Anna7777 View Post
    Is N dense in R?
    Let say (a,b)=(0,1) How come it does not contain an element of ℕ ?
    Is it because it's an open interval?
    Can someone please explain this to me? Also when would it be dense?
    Please understand that this is not a tutoring service. Also it is a fact that we have no idea upon what basis you are working.

    Usually the definition is that in the real numbers if $A\subset B$ then $A$bis dense in $B$ means between any two members of $B$ there is an element of $A$.

    As a basis for this study you should have worked out the properties of the integers.
    This is the one you need for this question: if $x\in\mathbb{R}^+$ then then $\exists\left\lfloor x \right\rfloor\in\mathbb{N}$ such that $\left\lfloor x \right\rfloor\le x<\left\lfloor x \right\rfloor+1$; i.e. There is no integer between $n~\&~n+1$

    Thus there can be no integer between $0~\&~1$.

    I hope you see that there is no simple explanation?
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    Re: Is N dense in R?

    Are you saying that you do not know what "(0, 1)" means? If so where did you see that notation? If you do know what it means, why is there any question if there are or are not integers in (0, 1)? What integers are you concerned might be in (0, 1)?
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    Re: Is N dense in R?

    I may not realize a probably simple thing but I don't see why do assume it is integer? Couldn't it be a rational number between (0,1)?
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    Re: Is N dense in R?

    Quote Originally Posted by Anna7777 View Post
    I may not realize a probably simple thing but I don't see why do assume it is integer? Couldn't it be a rational number between (0,1)?
    You are asking about \mathbb{N}. You need to look for an element n \in \mathbb{N} in (0, 1). There is no such number n in this interval.

    -Dan
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    Re: Is N dense in R?

    Quote Originally Posted by Anna7777 View Post
    I may not realize a probably simple thing but I don't see why do assume it is integer? Couldn't it be a rational number between (0,1)?
    Because that is what "N" means- the set of positive integers!
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