problem 2 25 points we wish to investigate a new property of relations

Question

Problem 2. (25 points) We wish to investigate a new property of relations which we will call
Lottian. A relation R on a set A is Lottian if:
Va, b, c € A: bRa A cRa → bRe
a) Below is a supposed proof that any Lottian relation is reflexive. Explain any and all issues with
the proof:
'Let R be a Lottian relation on the set A. Let z be an arbitrary element of A. We need to show
that Rr. Let y be any element in A such that Ry. Then we know that RyATRy is true. Thus
since R is Lottian, R is also true as desired.'
b) Construct an example of a relation that is Lottian but not reflexive.
c) Prove that if a relation R is reflexive and Lottian then R is an equivalence relation.