Answer to Question #334772 in Discrete Mathematics for zyzz

Question

Which of these relations on the set of all people are equivalence relations? Determine the properties of an equivalence relation that the others lack.

2a) {(a, b) ∣ a and b are the same age}

2b) {(a, b) ∣ a and b speak a common language}

Answer 1

Equivalence relations are one which satisfy

i) Reflexive i.e. (x,x) belong to R for all x

ii) Symmetric if (x,y) is in R, then (y,x) also would be in R

iii) Transitive (x,y) and (y,z) imply (x,z)

a) {(a, b) | a and b are the same age}.

This is equivalence since all conditions are satisfied.

b. {(a, b) | a and b speak a common language}.

Not equivalence suppose a,b speak common English, b and c speak common french then a,c may not have common language.

Transitive is in a and it lack in b.