Interactive Real Analysis
- part of Examples 2.1.2:
Find the bijections to prove the following statements:
Back
In each of the three cases we have to find a bijection between the two pairs of sets. - Let E be the set of all even integers, O be the set of odd integers. Then card(E) = card(O)
- Let E be the set of even integers, Z be the set of all integers. Then card(E) = card(Z)
- Let N be the set of natural numbers, Z be the set of all integers. Then card(N) = card(Z)
Back
- 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).
- 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).
- 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).