Question

a binary relation r on a set s is an equivalence relation if and only if it satisfies the three properties: (choose all that apply)

Answer 1

A binary relation r on a set s is considered an equivalence relation if it is reflexive, symmetric, and transitive. These necessary properties define the relation as equivalence.

Step-by-step explanation:

A binary relation r on a set s is called an equivalence relation if it satisfies the following three properties:

1. Reflexivity: Every element is related to itself. Formally, for every a in s, the relation r(a, a) holds.
2. Symmetry: If an element is related to another, then that second element is related to the first. In other words, if r(a, b) then r(b, a) for any a, b in s.
3. Transitivity: If an element is related to a second, and the second is related to a third, then the first element is related to the third. This means if r(a, b) and r(b, c), then r(a, c) for any a, b, c in s.

Note that these properties must apply to all elements of the set s for r to be considered an equivalence relation.