Cardinality of set of even numbers

[amit bhardwaj]

Is the cardinality of set of even numbers and set of natural numbers same?If no, then what is the cardinality of even numbers?

[Kapil Gupta]

Definitely it is same, as we are able to establish a one to one correspondence between Set of even numbers & set of Natural Numbers

[amit bhardwaj]

How and by which function we can establish one to one correspondence??

[Kapil Gupta]

Is the cardinality of set of even numbers and set of natural numbers same?If no, then what is the cardinality of even numbers?

Yes

Let there is a function f: E -> N such that f(e) = e/2
Now we have to prove that f is one-one and onto
First we will prove that f is one one
we prove it by contradiction,
if possible then assume f is not one one
there there must exist x1, x2 \in E and x1 != x2     (2)
but f(x1) = f(x2)                                    (1)
By definition of E, x1 = 2 x n1 and x2 = 2 x n2, where n1, n2 \in N
from (1)
2 x n1 = 2 x n2 => 2 x n1/2 = 2 x n2/2 (Halving both sides maintains equality)
it means n1 = n2. thus contradiction arrives, hence our assumption is wrong. Hence f:E->N is one one.
Now we prove that f is onto. Again proving by contradiction.
 Let f be not a onto function from E -> N
Let there is y \in N such that inverse_f(y) = 2 x y for y\in N 2y \notin E, which is impossible as per the definition of E

Hence there exists one to one onto function f: E->N as defined

[amit bhardwaj]

But sir, according to this function f(-2) wont be in the range and so it wont be a function!!!

[Kapil Gupta]

Agreed. Amit now I want you to solve this problem. I think you can write a good proof.


[amit bhardwaj]

Ok sir i shall try.

[amit bhardwaj]

Show if cardinality of set of even numbers and natural numbers is same?

[amit bhardwaj]

Sir is it necessary to show a bijective function exists between naturals and any other set to show their cardinality is equal ??

[Kapil Gupta]

There could be other methods as we did in case of prime numbers.

You may also try to argue in english that the way you are trying to map will definitely give one to one and onto mapping.