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  3. 2. let d be a relation on the natural numbers n de

2. Let D be a relation on the natural numbers N de?ned by D = {(m,n)...

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Advance Math

2. Let D be a relation on the natural numbers N de?ned by D ={(m,n) : m | n} (i.e., D(m,n) is true when n is divisible by m. Forthis problem, you’ll be proving that D is a partial order. Thismeans that you’ll need to prove that it is re?exive,anti-symmetric, and transitive.

(a) Prove that D is re?exive. (Yes, you already did this problemon one of the minihomework assignments. You don’t have to redo theproblem, but you should at least copy over your answer from thatassignments to this one.)

(b) Prove that D is anti-symmetric. You may use the followingfact: for any two natural numbers m and n, if m·n = 1, then m = 1and n = 1. (Note that D is only anti-symmetric because the domainis the natural numbers. If we switched to the domain of integers,then things would be completely di?erent.)

(c) Prove that D is transitive. (You’ve probably done a problemalready that is almost exactly the same as this.)

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