# Relations Help

• May 2nd 2009, 04:44 AM
grandstand
Relations Help
Hi, i've been mulling over this question for a few days now and i've got an answer, but it seems too simple to be correct.

8 = for all
R is relation
R is useful iff:
(u1) (8x¬(xRx)
(u2) 8xyz((xRy ^ yRz) implies (xRz))
(u3) 8xyz(xRy implies (xRz OR zRy))

Consider the relation ‘greater than’ on Natural numbers. Is it useful? Good? Cool?

I just need a true or false for each, but they all seem to be true (there are several more examples other than these three). I think i'm going about it the wrong way, and any advice to kick me off would be much apprieciated.
• May 2nd 2009, 05:33 AM
Plato
The relation $\mathcal{R}$ on $\mathbb{N}$ by $x\mathcal{R}y \text{ if and only if } x>y$.

$\left( {\forall x} \right)\left[ {x \not> x} \right]$.

Clearly $>$ is transitive. That is the second property.

For the third property suppose $\left\{ {x,y,z} \right\} \subset \mathbb{N}\;\& \; x\mathcal{R}y$. This means that $x>y$.
If $z \geqslant x\; \Rightarrow \;z > y\; \Rightarrow \; z\mathcal{R}y$
On the other hand $z < x\; \Rightarrow \;x\mathcal{R}z$.
• May 2nd 2009, 06:04 AM
grandstand
Hi, thanks for a quick response.

I'm understand what you have written, but i'm still not entirley clear on how to gauge if each property is true or false. This is the full part of that question:

call a relation R useful iff it has the following properties: Then the listed properties.

With R being >. As far as i can tell all those properties have to be true when used with the > (and the <). Maybe i'm just overthinking it and all the 5 properties in the question are true.
• May 2nd 2009, 06:51 AM
Plato
Quote:

Originally Posted by grandstand
Maybe i'm just overthinking it and all the 5 properties in the question are true.

What five properties?
You only gave three.
Which is it?
• May 2nd 2009, 06:56 AM
grandstand
There are 5 overall in the question.

Let us call a relation R useful iff it has the following properties:
(u1) 8x¬(xRx)
(u2) 8xyz((xRy ^ yRz) implies (xRz))
(u3) 8xyz(xRy ! (xRz OR zRy))
We call a relation R good iff it has the following properties:
(g1) 8x¬(xRx) (same as u1)
(g2) 8xyvw((xRy ^ vRw) implies (xRw OR vRy))
We call a relation R cool iff it has the following properties:
(r1) 8x¬(xRx) (same as u1)
(r2) 8xyvw((xRy ^ vRw) implies (xRw OR vRy)) (same as g2)
(r3) 8xyvz((xRy ^ yRz) implies (xRv OR vRz))

The question askes where R = > is useful, good and/or cool, but then to say which properties are true or false.