I need to prove that a relation is transitive.

Question

I got $(x,y)R(u,v) \Leftrightarrow x + v = y + u$

I have to prove that this is a transitive relation. We did not do any examples how to do this at school so as far as I came was:

$(x,y)R(a,b) \wedge (a,b)R(u,v) \Rightarrow (x,y)R(u,v)$

how do I go on from this point?

Answer 1

Suppose that $(x,y)\mathcal{R}(a,b)$ and also that $(a,b)\mathcal{R}(u,v)$. We wish to show that this implies that $(x,y)\mathcal{R}(u,v)$.

Since $(x,y)\mathcal{R}(a,b)$, by our definition of $\mathcal{R}$ that means that $x+b=y+a$.

Since $(a,b)\mathcal{R}(u,v)$, by our definition of $\mathcal{R}$ that means that $a+v=b+u$.

Now, try to combine those two equations in a convenient way to make it look at the end like $x+v=y+u$. Remember that this is what the final line should look like. Do not start with this as your first line.

Hint: What does $x+b+a+v$ look like?

Answer 2

You can't just write the implication. If that was the case you could just right that and you would be done.

You need to suppose that $(x,y) R (a,b)$ and $(a,b)R(u,v)$ then use the definitions of the relation to prove it.

$(x,y) R (a,b) \implies x+b=y+a$

$(a,b)R(u,v) \implies a+v=b+u$

you want to show $x+v=y+u$ now using the last two lines above can you do this. It shouldn't be too trick so I'll let you finish.

Also type in LaTeX much easier to read.