The discussion revolves around identifying examples of uncountable infinite sets that are not continuous. Participants are exploring definitions and characteristics of continuous and discrete sets in the context of set theory and probability.
The discussion is ongoing, with participants sharing definitions and examples while expressing confusion about the relationship between discrete and continuous sets. Some guidance has been offered regarding the nature of subsets and cardinality, but no consensus has been reached.
Participants are considering the implications of definitions from probability theory, specifically regarding discrete and continuous random variables, and are noting that examples may not be readily available in their reference materials.
jashua said:Can you give some examples of the infinite sets that are uncountable and that are not continuous?
I know the infinite sets that are countable and discrete, and I know the continuous sets, but couldn't find an example for the above situation.
jashua said:Thank you for your reply.
A set is called continuous if it contains an infinite number of elements equal to the number of points on a line segment.
A set is called discrete if it contains a finite number of elements or an unending sequence of elements with as many elements as there are whole numbers.
jashua said:I'm thinking about your example :) The number of all the subsets are infinite. There are also infinite number of discrete subsets and infinite number of continuous subsets. And then how do we deduce that the overall set is not continuous ? Is the reason that we have infinite number of discrete subsets?
I'm still confused. In fact, the question is related to the discrete/continuos random variables in probability theory. The question can be restated as: Is there a sample space which is not continuos and consists of uncountable infinite number of samples? Childer's Probability Theory book says yes, and it is beyond the scope of the book. However, it does not give an example.
I guess by "infinite number equal to..." you mean that the two sets are of the same Cardinality.A set is called continuous if it contains an infinite number of elements equal to the number of points on a line segment.