Final answer:
The relation R is an equivalence relation on the set of all natural numbers because it satisfies the reflexive, symmetric, and transitive properties. Each element is related to itself, the relation is symmetric as the order of elements does not affect the result, and it is transitive because the divisibility by 5 is consistent through the relations.
Step-by-step explanation:
To prove that the relation R is an equivalence relation on the set of all natural numbers, we need to show that R is reflexive, symmetric, and transitive.
Reflexive:
For all x in natural numbers, (x, x) is in R because the difference x - x is 0, which is divisible by 5. This shows R is reflexive.
Symmetric:
For any (x, y) in R, it implies that the difference x - y is divisible by 5. If x - y is divisible by 5, so is y - x because the divisibility by 5 does not depend on the order of subtraction. Hence, (y, x) is also in R, proving that R is symmetric.
Transitive:
If we have two relations (x, y) and (y, z) in R, this means x - y and y - z are both divisible by 5. To find if (x, z) is in R, we consider the sum (x - y) + (y - z) = x - z, which is also divisible by 5. Hence, (x, z) is in R, demonstrating that R is transitive.
Since R is reflexive, symmetric, and transitive, we conclude that R is an equivalence relation on the set of all natural numbers.