# Interactive Real Analysis

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## 2.1. Countable Infinity

### Examples 2.1.2:

Find the bijections to prove the following statements:
1. Let E be the set of all even integers, O be the set of odd integers. Then card(E) = card(O)
2. Let E be the set of even integers, Z be the set of all integers. Then card(E) = card(Z)
3. Let N be the set of natural numbers, Z be the set of all integers. Then card(N) = card(Z)
In each of the three cases we have to find a bijection between the two pairs of sets.

1. Define the function f(n) = n + 1 with domain E and range O. Then the function f is clearly one-to-one and onto, hence it is a bijection. Now f is a bijection between E and O, so that card(E) = card(O).

2. Define the function f(n) = 2n with domain Z and range E. Then it is straight-forward to show that this function is one-to-one and onto, giving the required bijection. Hence, card(Z) = card(E).

3. Define the following function: f(n) = n / 2 if n is even and f(n) =- (n-1) / 2 if n is odd, with domain N and range Z. Again, it is not hard to show that this function is one-to-one and onto, and therefore card(N) = card(Z).
The actual details of proving that the functions are bijections are left as an exercise.
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