Answer to Question #345316 in Discrete Mathematics for Fahmina

Question #345316

Let β€˜R’ be a relation defined on a set of integers Z as follows:



βˆ€ π‘Ž, 𝑏 ∈ 𝑍, π‘Žπ‘…π‘ iff 𝑏 = a^r



for some integer π‘Ÿ. Show that R is a partially



ordered relation.




1
Expert's answer
2022-06-01T13:45:02-0400

"R" is a partially ordered relation if and only if it is reflexive, antisymmetric and transitive.


1) reflexive

"aRa," because "a=a^1."


2) antisymmetric

If "aRb" and "bRa," then "b=a^m" and "a=b^n" for some integers "m" and "n."Β 

So, "b=a^m=(b^n)^m=b^{nm}" and it means that "mn=1."

There are two cases: "m=n=1" and "m=n=-1."

If "m=n=1," then "a=b."

If "m=n=-1," then it is possible only for "a=b=\\pm1."


3) transitive

If "aRb" and "bRc," then "b=a^m" and "c=b^n" for some integers "m" and "n."

Since "c=b^n=(a^m)^n=a^{mn}," we have that "aRc."


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