Let e stand for the set of all even natural numbers (so e={2,4,6,} and o stand for the set of all odd natural numbers (so o={1,3,5,}). show that the sets e and o have the same cardinality by describing an explicit one-to-one correspondence between the two sets.

Cardinality is the number of elements in a set. Both of the sets E and O have infinite elements, but there are different types and sizes of infinity. Luckily, all you have to do to prove that two sets have the same cardinality is describe an explicit one-to-one correspondence between the two.
For odds and evens, this is a relatively easy thing to do - all odds are just one less than an even. So, the function that relates the two sets is
f(n) = n + 1
or, depending on how you write it,
E(n) = O(n) + 1