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 can select a subset that contains a finite number of elements. Let's choose X = {1, 2}.
Since X has only two elements, it is clear that there is no one-to-one correspondence between X and N. This is because N contains an infinite number of elements (natural numbers), while X contains only two. Therefore, X is a proper subset of N without a one-to-one correspondence.
To find a proper subset Y of N such that there is a one-to-one correspondence between Y and N, we need to select a subset that contains an infinite number of elements. Let's choose Y = N itself.
Since N is the set of natural numbers, it contains an infinite number of elements, and each natural number can be mapped to itself in a one-to-one correspondence. For example, 1 can be mapped to 1, 2 to 2, 3 to 3, and so on. Therefore, Y = N is a proper subset of N with a one-to-one correspondence.