One-to-one correspondence?

Question

1. Is there a 1-1-correspondence between the set of irrational nos. and the natural nos.? if so, how?

2. How would u construct a 1-1-c between Natural and Real nos?

But my teacher said that we could construct a 1-1-c between natural and real. We just have to prove/construct but I can't find how.. I am really confused, any site eh?

My bad, NATURAL and RATIONAL i mean.. Sorry, I had a hard time dealing with letters to represent sets, I was mixed up.

Answer 1

1. No, there is not. Between each pair of natural numbers (no matter how close they are to one another) the is an infinite number of irrational numbers. Therefore, the irrationals are of a higher order of infinity than are the rationals.

2. It's hard to describe in words, but there is a method involving the laying out of a table like a cross-reference table. I'm sure you could find it in a book on it.

There is a good description at this site:
http://www.personal.psu.edu/faculty/
j/x/jxt18/Math21_WEB/Lesson1/
lesson_one.htm

Hope this helps.

Answer 2

1. The cardinality of the set of irrational numbers is c (the cardinal of the continuum), and the cardinality of the natural numbers is aleph naught, so there cannot be a 1 - 1 correspondence between these two sets.

2. The same disparity exists between the cardinalities here, so there cannot be a 1 - 1 correspondence.

Answer 3

No. The natural numbers and the rational numbers can be set up in one-to-one correspondence, but not between natural numbers and irrational numbers, or between natural numbers and real numbers.
.

Answer 4

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Answer 5

oouch! THAT hurts my head~ (c: Maybe we can PrEtEnD that I do ~ NOT~ know what you are asking
Are we talking GOOGLEPLEX x's 14 or something or just emails and letters sent to others?

Answer 6

I've no idea, I knew I should have did my homework and actually learnt something at school !