







Countable and uncountable sets? 
20160115 

From wilson: what are the countable and uncountable sets? Answered by Penny Nom. 





Cantor's diagonal argument 
20080126 

From David: Cantor's theory using a diagonal across a list of real numbers to proven uncoutability has always puzzled me.First in base ten, it feels like hocus pocus so I began thinking of the Boolean numbers as truer representations of place value (on,off). Secondly his list was always arbitrary or so I recollect. Therefore, I suggested using a seriesA=.10000....,B=.01000. C=.11000, etc.
Any diagonal is already located among the numbers listed. My only alteration is that since the final digit is always unrepresentably either one or zero, but it must be one or the other, I make an assumption that if x= .abc...1 and y= .abc...2 the only two possibilities and I choose to count F=x+y then then the numbers are countable= Z=sumFi,where I=2+2^2+2^3...
I hope this sketch is enough description, I asked Rudy Rucker more formally but got no mathematical response, someone else gave me some tale about slippery epsilon. What do tyou think of recasting his proofs in more rigorous form?
David French Answered by Claude Tardif and Walter Whiteley. 





More on the cardinality of sets 
20070727 

From Mac: Can you please help me to find and verify whether the following are
finite, countably infinite and uncountable ? Answered by Harley Weston. 





Countable and uncountable sets 
20070724 

From Mac: Hi, i tried to read few webpages related to the countably infinite and uncountable sets.
Even i read few questions from this forum.
But i am not convinced with this explanation. If you have any good book that
explains this in layman term, please redirect me to that.
1) Can you please explain what is the difference between these too ?
2) How could you say set of Natural number and set of even numbers are countably
infinite ?
N={1,2,3,...} and Even= {2,4,6,...}
When an element in the even set is some 2n, we will map it to 'n'.So
now we have a bigger number(2n) right ?
Sorry, i didn't understand that.
...
Can you please help me out to understand that ? Answered by Harley Weston. 





Countable and uncountable sets 
20070213 

From piyush: we se that union of countably infinite no of sets having countably infinite number of elements is a countable set we can express p(n) (i.e power set of natural number) as a union of countable infinite number of sets i.e p(n)=s1Us2Us3..... where s1=null s2={1,2,3,4,5..........} s3={{1,1},{1,2},{1,3},..............{2,1},{2,2}........} using the same statement can we prove that power set of natural number is a infinit countable set Answered by Penny Nom and Claude Tardif. 





The real numbers with decimal representations consisting of all 1s. 
20061029 

From Ivessa: Determine if the following set is countable or uncountable : the real numbers with decimal representations consisting of all 1s. Answered by Steve La Rocque and Walter Whiteley. 





The cartesian product of a countably infinite collection of countably infinite sets 
20060325 

From Geetha: Is the cartesian product of a countably infinite collection of countably infinite sets countable infinite? Answered by Penny Nom. 





Cardinality of sets 
20011119 

From Tania:
 Show that the cardinality of P(X) (the power set of X) is equal to the cardinality of the set of all functions from X into {0,1}.
 Show that (the cardinality of the natural numbers set) N = NxNxN.
 Show that the cardinality of the set of prime numbers is the same as the cardinality of N+
Answered by Walter Whiteley. 





Subsets of a countably infinite set 
20011114 

From Tania: How could I show (and explain to my son) that any countably infinite set has uncontably many infinite subsets of which any two have only a finite number of elements in common? Answered by Claude Tardif. 

