Two questions: (1) Show that N X N can be written as an infinite union ofdisjoin

Question

Two questions:

(1) Show that N X N can be written as an infinite union ofdisjoint infinite sets (where N are the real numbers and NXN is thecartesian product of the reals)

(2) Show there is a bijection from RXR ----> R (where R isthe real numbers).

Any help would be greatly appreciated. Will rate.

Explanation / Answer

1) I think another way to say what you want is: find a cover ofRxR, which is not to hard: Ui[(ni,n+1i)x(ni,n+1i)] where the i isan index from the integers. 2) the diagonal of R x R is defined as the line: {(x,x)| x is inR}. then the map (x,x)-->x should be sufficient: the map is well defined: assume that one element of the domain gets mapped to 2 elements ofthe range: (x,x)->x1 and (x,x)->x2. Then by def,(x,x)=(x1,x1). similarly, (x,x)=(x2,x2)..by the transitivity ofequality, x1=x2. So the map is well defined the map is one to one: assume that 2 elements in the domain map to one element in therange, (x1,x1)->x and (x2,x2)->x. Again by transitivity,(x1,x1)=(x2,x2) the map is onto this is just about obvious by defintion almost. take an element inthe range x and show that there exists an element in the domainthat maps to it. well if x is in R then (x,x)is in RxR. So(x,x)-->x. hope that helps!

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