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Let Q be the graph consisting of vertices and edges of a 3-dimensional cube. Two relations are defined on the vertices of Q.

• R1={(v,w):the shortest path from v to w has an odd number of edges}.

• R2={(v,w):the shortest path from v to w has an even number of edges.

a) Exactly one of R1 and R2 is an equivalence relation

b) Both R1 and R2 are equivalence relations

c) None of R1 and R2 is an equivalence relation

Thank you very much.

Note: this is a revised answer

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