Exercise 1) Prove that “infinity type 1 divided by two is still infinity type 1”. Do this by realizing that the set of even numbers is a meaningful way of describing “infinity type 1 divided by two”, since we can split evenly into odd and even parts. Then define a bijective function from to , thus showing that A and B have the same cardinality!
Solution: We do just what the hint suggests. Let the set A and B be as the hint describes, and define a function from A to B simply by sending each element in A to “2 times itself”. Namely, we send 1 in A to 2 in B, 2 in A to 4 in B, 3 in A to 6 in B, 4 in A to 8 in B, and so on, forever. This is clearly injective, because if two elements in A are sent to the same element in B, then that means that those two elements “times 2” are equal to each other. But if “something” times two is equal to “something else” times two, then “something” must be equal to “something else”. Moreover, this function is surjective since each element N in B is hit precisely by “N divided by 2” in A. Thus this function is bijective, and we’ve shown that “infinity type 1 divided by 2 is infinity type 1”!