Let (B, 3) be a partially ordered set, A be a set, and f: A B...

 

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Let (B, 3) be a partially ordered set, A be a set, and f: A B a function from A to B. Define RCA x A as follows: (a,b) = R if and only if f(a) f(b) (a) Give a counterexample to show that in general R is not a partial order. (b) (i) State a restriction on f that will ensure R is a partial order, and (ii) Prove that under that restriction R is a partial order. (c) Prove that RnR is an equivalence relation. (4 marks) (1 mark) (9 marks) (6 marks) Let (B, 3) be a partially ordered set, A be a set, and f: A B a function from A to B. Define RCA x A as follows: (a,b) = R if and only if f(a) f(b) (a) Give a counterexample to show that in general R is not a partial order. (b) (i) State a restriction on f that will ensure R is a partial order, and (ii) Prove that under that restriction R is a partial order. (c) Prove that RnR is an equivalence relation. (4 marks) (1 mark) (9 marks) (6 marks)