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Math Help - Definition clarification - Countable infinite

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    Definition clarification - Countable infinite

    I was reading up on discrete random variables, when I came across the definition:

    A random variable is called discrete if its range (the set of values that it can take) is finite or at most countably infinite. A random variable that can take an uncountably infinite number of values is not discrete. For an example, consider the experiment of choosing a point a from the interval [−1, 1]. The random variable that associates the numerical value a^2 to the outcome a is not discrete.

    I do not quite understand the meaning, even after looking at other definitions which talked about a 1:1 correspondence with positive integers.

    Does this mean that (1,2], [0,3] are countably infinite, whereas [-2,2] or (-3,2] are not, as there are frequently two numbers in the set which will correspond to the same positive integer? (eg -1.5^2 and 1.5^2 both correspond to 2.25)?

    additionally,
    What does the notation sgn(a) = { 1 if a>0, 0 if a=0, -1 if a<0
    imply? I have not seen 'sgn' before

    thanks
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    Re: Definition clarification - Countable infinite

    Quote Originally Posted by 99.95 View Post
    A random variable is called discrete if its range (the set of values that it can take) is finite or at most countably infinite. A random variable that can take an uncountably infinite number of values is not discrete. For an example, consider the experiment of choosing a point a from the interval [−1, 1]. The random variable that associates the numerical value a^2 to the outcome a is not discrete.
    One of Georg Cantor's many contributions was to show (0,1) is an uncountable set using his famous diagonal argument.

    If a<b then (a,b) is an open interval and a+t(b-a),~t\in(0,1) maps (0,1)\to (a,b) one-to-one and onto.

    Therefore, any set of real numbers which contains any open interval most be uncountable.

    Because (0,1)\subset [-1,2) the set [-1,2) most be uncountable.

    Be careful. Any set of real numbers that is not countable most be uncountable.
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    Re: Definition clarification - Countable infinite

    sgn(a) = { 1 if a>0, 0 if a=0, -1 if a<0
    is the "sign" function so called because it returns "+1" for any positive number, "-1" for any negative number, 0 for 0 which is neither positive nor negative.
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