# Thread: Determine what kind of relations these are

1. ## Determine what kind of relations these are

Consider a relation $R$ defined on the integers. Determine for the following if the relations are reflexive, symmetric, antisymmetric, transitive, partial orders, equivalence relations.

a) $R=\{(a,b)|a^2\geq b\}$

b) $R=\{(a,b)|a=|b|\}$

c) $R=\{(a,b)|5a=2b\}$

d) $R=\{(a,b)|ab\leq0\}$

2. Originally Posted by Runty
Consider a relation $R$ defined on the integers. Determine for the following if the relations are reflexive, symmetric, antisymmetric, transitive, partial orders, equivalence relations.

a) $R=\{(a,b)|a^2\geq b\}$

b) $R=\{(a,b)|a=|b|\}$

c) $R=\{(a,b)|5a=2b\}$

d) $R=\{(a,b)|ab\leq0\}$
Where is your work towards the solutions?
What have you done?

3. I've done some work on these for now (I don't know how to determine partial orders or equivalence relations), though I'm not sure if all my current answers are correct.

I'm unfortunately not able to show my work for my answers, as it would take a very long time to write it all out in Latex. I will, however, write out the conditions for each property (reflexive, symmetric, anti-symmetric, transitive) if it is true for the relation. For these, the sets are all integers (basically, $x\in Z$).

Reflexive: $\forall x\in Z, (x,x)\in R$
Symmetric: $\forall x,y\in Z, (x,y)\in R\wedge(y,x)\in R$
Anti-Symmetric: $\forall x,y\in Z, ((x,y)\in R\wedge(y,x)\in R)\rightarrow x=y$
Transitive: $\forall x,y,z\in Z, ((x,y)\in R\wedge(y,z)\in R)\rightarrow (x,z)\in R$

a) $R=\{(a,b)|a^2\geq b\}$
Reflexive, not symmetric, not anti-symmetric, transitive

b) $R=\{(a,b)|a=|b|\}$
Not reflexive, not symmetric, anti-symmetric, transitive

c) $R=\{(a,b)|5a=2b\}$
Not reflexive, not symmetric, anti-symmetric, not transitive

d) $R=\{(a,b)|ab\leq0\}$
Not reflexive, symmetric, not anti-symmetric, not transitive

Can anyone help me with determining if these are partial orders relations and/or equivalence relations? I'm afraid I don't know how to do those.

EDIT: One more thing. Plato, just so I'm clear, is there some unwritten rule about us having to do some work on our questions before asking them on the forum? If so, I'll accept that. I just want to make sure.

4. Originally Posted by Runty
. Plato, just so I'm clear, is there some unwritten rule about us having to do some work on our questions before asking them on the forum? If so, I'll accept that. I just want to make sure.
Well it is clear from the title of this forum: Math Help forum.
We are not a homework service.
If you only post a list of problems then what are we to think?
Without some guidance from you, by not seeing what you understand, how can we help?

Originally Posted by Runty
I've done some work on these for now (I don't know how to determine partial orders or equivalence relations).
Can anyone help me with determining if these are partial orders relations and/or equivalence relations?
Any relation on a set that is reflexive, anti-symmetric, and transitive is a partial ordering.

Any relation on a set that is reflexive, symmetric, and transitive is an equivalence relation .

5. Thanks for that bit on partial orders and equivalence relations.

Now, concerning the answers I provided earlier, did I make a mistake anywhere? I'm particularly concerned about the second one, which contains an absolute value part (I always get things messed up when absolute value brackets are involved). I'm not 100-percent sure if the second one is anti-symmetric. Besides that, I'm not entirely confident with my answers.

6. The part b) is correct.
I looked over the others and saw no errors.

7. Hey guys,

Isn't the relation in part a not transitive? The pairs (2,3), and (3,5) are in the relation but (2,5) is not.

Cheers

8. Originally Posted by Lexa
Hey guys,

Isn't the relation in part a not transitive? The pairs (2,3), and (3,5) are in the relation but (2,5) is not.

Cheers
Thanks for pointing that out. I didn't catch that earlier.

9. You're welcome, glad to help.