Problem 1 (22 pts.) Expected Time: 1-2 hours. Similar to: Tutorial 10 Problems 1-3, Claims in Properties of Binary Relations lecture Type of Practice: Modify Existing Proof to Prove a Similar Claim. This problem asks you to con- struct direct proofs by modifying the proof of a very similar result. Claim. Let G = (V, E) be any undirected graph, and let f : V - {1, 2, ..., k} be a k-colouring of G, where k = x(G). The binary relation =f on V defined by uf v if and only if f(u) = f(v) is an equivalence relation. This problem will guide you through the proof of this claim and a bit more. At a minimum you should have as many rows as the given tables, but you may add more if you desire. Note: Some nice variables to use for arbitrary vertices are u, v, w. (a) (4 pts.) Fill in the following table with a direct proof that =y is reflexive. Mathematical Statement Reason this Statement is True From the Approved List) WTS by def of reflexive Consider any by rephrasing WTS (b) (6 pts.) Fill in the following table with a direct proof that = is symmetric. Mathematical Statement Reason this Statement is True (From the Approved List) WTS by def of symmetric Consider any this is the antecedent Suppose WTS this is the consequent => 3