Let (X, lessthanorequalto), (Y, precedsorequalto) be ordered sets. We say that t

Question

Let (X, lessthanorequalto), (Y, precedsorequalto) be ordered sets. We say that they are isomorphic (meaning that they "look the same" from the point of view of ordering) if there exists a bijection f:X rightarrow Y such that for every x, y elementof X, we have x lessthanorequalto y if and only if f(x) precedsorequalto f(y). Draw Hasse diagrams for all nonisomorphic 3-element posets. Prove that any two n-element linearly ordered sets are isomorphic. Find two nonisomorphic linear orderings of the set of all natural numbers. Can you find infinitely many nonisomorphic linear orderings of N? Uncountably many (for readers knowing something about the cardinalities of infinite sets)?

Explanation / Answer

a,b, c are 3 elements

a <= b <=c

a<=b,c

a,b <= c

a<=b , c unrelated

a,b,c (not related)

5 different sets

b)

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