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

Re: Definition clarification - Countable infinite

Quote:

Originally Posted by

**99.95** 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 $\displaystyle (0,1)$ is an uncountable set using his famous *diagonal argument*.

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

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

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

Be careful. Any set of real numbers that is not countable most be uncountable.

Re: Definition clarification - Countable infinite

Quote:

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.