Let R1 be a binary relation on the set of integers defined as follows:
R1 = {(x, y) / 4 divides x – y}. Determine whether the given relation R1 is an equivalence relation on the set {1, 2, 3, 4, 5}.
To determine whether the relation R1 is an equivalence relation on the set {1, 2, 3, 4, 5}, we need to check if it satisfies three properties: reflexivity, symmetry, and transitivity.
1. Reflexivity: A relation is reflexive if every element is related to itself. In our case, we need to check if (x, x) is in R1 for every x in the set {1, 2, 3, 4, 5}.
To check reflexivity, we need to find out if 4 divides x - x (which simplifies to 0), for every x in the given set. Simplifying further, this means we need to check if 4 divides 0 for every x in the set. Since 4 divides 0 (since any number divided by 0 is 0), reflexivity is satisfied.
2. Symmetry: A relation is symmetric if (x, y) is in the relation whenever (y, x) is in the relation. In our case, we need to check if whenever (x, y) is in R1, then (y, x) is also in R1.
To check symmetry, we need to verify if for every (x, y) in R1, we can also find (y, x) in R1. This means checking if 4 divides x - y implies that 4 divides y - x. If 4 divides x - y, this implies that x - y is a multiple of 4. And since the negative of a multiple of 4 is also a multiple of 4, y - x is also divisible by 4. Therefore, symmetry is satisfied.
3. Transitivity: A relation is transitive if (x, y) and (y,z) are in the relation, then (x, z) is also in the relation. In our case, we need to check if whenever (x, y) and (y, z) are in R1, then (x, z) is also in R1.
To check transitivity, we need to verify if for every (x, y) and (y, z) in R1, we can find (x, z) in R1. This means checking if 4 divides x - y and 4 divides y - z, it implies that 4 divides x - z. By subtracting the two given divisibility conditions, we find that 4 divides (x - y) + (y - z), which simplifies to 4 divides x - z. Therefore, transitivity is satisfied.
Since the relation R1 satisfies reflexivity, symmetry, and transitivity, we can conclude that R1 is an equivalence relation on the set {1, 2, 3, 4, 5}.