Let A = Z, the set of integers and let R is defined by a R b if and only if a ≤ b. Is R is an equivalence relation ?

No, R is not an equivalence relation.

For R to be an equivalence relation, it must satisfy three conditions: reflexivity, symmetry, and transitivity.

Reflexivity: For all a ∈ Z, a R a. This means that every integer must be related to itself. In this case, a ≤ a is always true, so reflexivity holds.

Symmetry: For all a, b ∈ Z, if a R b, then b R a. This means that if a is related to b, then b must also be related to a. However, this is not always true for R. For example, if a = 1 and b = 2, then a R b because 1 ≤ 2, but b is not related to a because 2 > 1. Therefore, symmetry does not hold for R.

Transitivity: For all a, b, c ∈ Z, if a R b and b R c, then a R c. This means that if a is related to b and b is related to c, then a must also be related to c. Transitivity holds for R because if a ≤ b and b ≤ c, then a ≤ c.

Since R does not satisfy the symmetry condition, it is not an equivalence relation.