(a) Let A be a finite set and let RAA be a relation. We say that R is negatively transitive whenever the following holds: x,y,zA,(x,y)/R(y,z)/R(x,z)/R Let us also say R is asymmetric whenever the following holds: x,yA,(x,y)R(y,x)/R 1. Prove that if R is both asymmetric and negatively transitive, then R is transitive. 2. Assume now that R is transitive. Is it the case that R is both asymmetric and negatively transitive? If so, prove it. Otherwise, give a counterexample. (b) Recall that a binary relation R on A is irreflexive if and only if, for every aA,(a,a)/R. A binary relation R on A is anti-transitive if: a,b,cA,(a,b)R(b,c)R(a,c)/R Show that an anti-transitive relation is irreflexive. (Hint: proceed by contradiction.)