example of uncountable family of subsets of a countable set with finite intersections
We wish to give an answer to the following:
Example. Let be a real number. Express using digits
where each . With we associate the following natural numbers
Now define (here stands for ,,the power set of ”) by
is injective. Indeed, note that for any if , then (this is because equal numbers have equal ,,length” and this is because each has at the begining, zeros are not the problem). Therefore, if for some , then it follows, that for each , but this implies that corresponding digits of and are equal. Thus .
This shows, that is an uncountable family of subsets of . Now in order to prove that is finite whenever it is enough to show that we can uniquely reconstruct from any infinite sequence of numbers from . This can be proved by using similar techniques as before and we leave it as a simple exercise.
|Title||example of uncountable family of subsets of a countable set with finite intersections|
|Date of creation||2013-03-22 19:16:29|
|Last modified on||2013-03-22 19:16:29|
|Last modified by||joking (16130)|