Define a relation R on N such that there is a one-to-one correspondence between R and N.
To define a relation R on N (the set of natural numbers) such that there is a one-to-one correspondence between R and N, we can use the concept of cardinality and the notion of a bijection.
A one-to-one correspondence, also known as a bijection, is a function between two sets where each element in the first set is related to exactly one element in the second set, and vice versa.
In this case, we want to define a relation R on N such that every element in N is related to exactly one element in R, and vice versa. Since N is an infinite set, we need to find a way to establish a one-to-one correspondence between the elements of N and the elements of R.
One approach is to define a mapping between the natural numbers and a subset of the natural numbers, such that every element in N is related to exactly one element in R. For example, we can define the following relation:
R = {(n, 2n) | n ∈ N}
In this relation, each natural number n is related to its double, 2n. For instance, (1, 2), (2, 4), (3, 6), and so on. This relation establishes a one-to-one correspondence between N and the subset of even numbers.
To understand why this relation establishes a one-to-one correspondence, we can see that for every natural number n, there is a unique corresponding element in R (its double, 2n). Likewise, for every element in R, there is a unique corresponding natural number (half of the value).
Note that this approach can be generalized to other subsets of N as well. For example, we could define a relation R = {(n, 3n) | n ∈ N}, where each natural number n is related to its triple, 3n, establishing a one-to-one correspondence between N and the subset of numbers that are multiples of 3.
In summary, to define a relation R on N such that there is a one-to-one correspondence with N, we can establish a mapping between the elements of N and a subset of N where each element is related to exactly one element in R.