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 need to establish a relationship that pairs each element in N with a unique element in R.
A common approach to achieve this is to define a relation R that maps each natural number n to itself, i.e., R = {(n, n) | n ∈ N}. This relation essentially pairs each natural number with itself, creating a one-to-one correspondence between N and R.
To see why this definition establishes a one-to-one correspondence, consider the following:
1. Every natural number n is included in the relation R: For every n ∈ N, the pair (n, n) is present in the relation R. This ensures that every natural number is accounted for in the correspondence.
2. Every pair in R is unique: Since each pair (n, n) maps a unique natural number to itself, there are no repeated or duplicated pairs in R. Thus, each element in R corresponds to a unique element in N.
By defining the relation R = {(n, n) | n ∈ N}, we establish a one-to-one correspondence between the set of natural numbers N and the relation R.