Show that R is a partial order on ℤ.

[s3a]

Homework Statement

The problem and solution are attached as TheProblemAndSolution.jpg.

I will also copy down the problem and solution here.:
Problem:
Consider the set ℤ of integers. Define aRb by b = a^r for some positive integer r. Show that R is a partial order on ℤ, that is, show that R is (a) reflexive; (b) antisymmetric; (c) transitive.

Solution:
(a) R is reflexive since a = a^1.

(b) Suppose aRb and bRa, say b = a^r and a = b^s. Then a = (a^r)^s = a^(rs).
There are three possibilities: (i) rs = 1, (ii) a = 1, and (iii) a = -1.
If rs = 1 then r = 1 and s = 1 and so a = b. If a = 1 then b = 1^r = 1 = a, and, similarly, if b = 1 then a = 1.
Lastly, if a = –1 then b = –1 (since b ≠ 1) and a = b. In all three cases, a = b.
Thus R is antisymmetric.

(c) Suppose aRb and bRc say b = a^r and c = b^s. Then c = (a^r)^s = a^(rs) and, therefore, aRc. Hence R is transitive.

Accordingly, R is a partial order on ℤ.

Homework Equations

Definitions of partial order, reflexive binary relations, antisymmetric binary relations, transitive binary relations.

The Attempt at a Solution

I don't get why if a = –1, then b = –1. If I assume that b ≠ 1, then I get it, but why is b = 1 forbidden?

Any input would be greatly appreciated!

 

[HallsofIvy]

If b= a "aRb= bRa" is trivial so you can assume b\ne a. Since this is the case a= 1, you may assume that b is not 1.

 

[Orodruin]

By your supposition, bRa, a = b^r for some r. This cannot be true if a = -1 and b = 1.

 

[s3a]

Sorry, I double-posted.

 

[s3a]

Orodruin, the thing is, if a ≠ b, then antisymmetry does not hold, and we can't assume that a = b for antisymmetry to hold, because we're trying to show that antisymmetry holds in the first place by showing that a = b must hold.

HallsofIvy, are you saying I should assume a ≠ b, and that I should then show that the equations don't always hold if a ≠ b?

I'm still confused about how I can say that –1 = (–1)^(positive_exponent), because positive_exponent must always be an odd integer for that to hold, but it could be an even integer too, so what do I do?

 

[s3a]

Sorry, I double-posted.

 

[s3a]

Never mind, I got it.

Because a = b^s implies that –1 = 1^s is false, b ≠ 1.

Thank you both.