Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) | a − b is even}, is an equivalence relation.

To demonstrate that the relation R defined on the set A = {1, 2, 3, 4, 5} by R = {(a, b) | a − b is even} is an equivalence relation, we need to verify three properties: reflexivity, symmetry, and transitivity.

Reflexivity

A relation is reflexive if every element is related to itself. For any element a in A:

• Calculate a - a = 0, which is even.

Since this holds for all a in A, R is reflexive.

Symmetry

A relation is symmetric if for any elements a and b in A, whenever (a, b) is in R, then (b, a) must also be in R. Assume (a, b) is in R:

• This means a - b is even.
• Then, b - a = -(a - b), which is also even.

Thus, if (a, b) is in R, (b, a) is also in R, confirming symmetry.

Transitivity

A relation is transitive if whenever (a, b) and (b, c) are in R, then (a, c) must also be in R. Assume (a, b) and (b, c) are in R:

• This means a - b is even and b - c is even.
• Adding these gives (a - b) + (b - c) = a - c, which is even.

Therefore, (a, c) is in R, proving transitivity.

Conclusion

Since R satisfies reflexivity, symmetry, and transitivity, we conclude that R is indeed an equivalence relation on the set A.