Let N= {1,2,3,...}, that is, N is the set of natural numbers. Find a proper subset X of N such that there is not a one-to-one correspondence between X and N. Find a proper subset Y of N such that there is a one-to-one correspondence between Y and N.
To find a proper subset X of N such that there is not a one-to-one correspondence between X and N, we need to find a subset of N that has fewer elements than N. One such subset is the set of even numbers: X = {2, 4, 6, ...}. We can see that X is a proper subset of N since it only contains the even numbers, and there is no one-to-one correspondence between X and N. This is because for every natural number n in N, we can find a unique even number 2n in X, but for every even number e in X, there is no corresponding natural number in N.
Now, to find a proper subset Y of N such that there is a one-to-one correspondence between Y and N, we need to find a subset of N that has the same cardinality (number of elements) as N. One such subset is the set of prime numbers: Y = {2, 3, 5, 7, 11, 13, ...}. We can see that Y is a proper subset of N since it only contains the prime numbers, and there is a one-to-one correspondence between Y and N. This is because every natural number n in N can be uniquely mapped to a prime number in Y, and every prime number p in Y can be uniquely mapped to a natural number in N by its position in the sequence of primes.
So, X = {2, 4, 6, ...} is a proper subset of N without a one-to-one correspondence, and Y = {2, 3, 5, 7, 11, 13, ...} is a proper subset of N with a one-to-one correspondence.