Algebra Help!!!

[LastPoet]

Okay, let me see if I can make this clear, found some trouble in an algebra book I'm reading, perhaps someone can explain how to solve that problem, this is in the Set Theory part, very basic, at the very start, I can't move forward until I get this one right. So, here it goes:

the book states:

A relation in a set A is defined by all the possible parts of A^2
beeing A^2 the cartesian product A*A, and parts of it all the possible subsets(including empty set and the same set)
Define relations of equivalence in a set with 2 elements, beeing a relation of equivalence part of A^2, that is defined by, symbolizing ? as such relation:
Reflexive: for all x that belongs to A: x?x
Symmetric: x?y -> y?x
Transitive: x?y, y?z -> x?z

I´m really boggled down on the transitive property on a set with 2 elements, according to this book...

[Tiberius]

I'm not entirely sure what your question is, but I'll assume you just don't understand what transitive relations are. In my experience, all set theory notes just make it far too confusing, for what is a mindbogglingly simple relation. I spend ages looking through notes trying to understand the set theory notation, and in the end it only made sense when I read a basic description online.

For simplicities sake, we'll have set A = {X, Y, Z}.

A relationship on A is transitive if whenever X is related to Y, and Y is related to Z, X is also related to Z. You can think about it in terms of traveling between cities. If each element is a city, and there is a road between X & Y, and a road between Y & Z, then you must be able to drive from X to Z.

A transitive relationship holds if this is true for every configuration of X, Y, and Z. So if Z -> X, and X -> Y, then Z must also go to Y for the relationship to be transitive.

Hope that helps.

[LastPoet]

Yeah, I understand transitivity, but the problem I'm having is establishing such relation in a set with 2 elements, eg. A={x,y}.

Indeed, this book is extremely hard to digest, almost all pages present 2-3 'demonstrate that', 'show how' questions.

[Tiberius]

Ok, sorry for the misunderstanding. In that case any relationship on a 2 element set would be transitive.

[LastPoet]

Ok, I get what you mean, I wasn´t abstracting the idea very well, I'll compose a demonstration, I'll present it when I'm done. Man, I love math, it makes you fight for understand

Thanks Adrian.

[smarika]

If X->Y, Z->X then we can say Y->Z.
This can be prove through mapping.
Here you are talking about the transitive relationship, according to it, transitive relationships must have binary element, therefore (X,Y), and (X,Z), there must exist (Y,Z) for it to be true.
I hope it will help you.