1. ## Relations help!!

Hey guys, i'm having some problems answering the last question of my assignment....so hopefully you can help

8. Test the relation R for the properties of reflexivity, symmetry, antisymmetry and transitivity if R is on the set of positive integers, and is given by (x,y)eR if x<=2y

*that "e" is supposed to be the, "in the set" character.

Hope to hear from you soon,

2. Hello, Storm20!

This is a tricky one . . .

8. Test the relation $\displaystyle R$ for the properties of reflexivity, symmetry, antisymmetry and transitivity
if $\displaystyle R$ is on the set of positive integers, and is given by: $\displaystyle (x,y) \in R\text{ if }x \leq 2y$

Reflexive: .$\displaystyle x R x$
. . $\displaystyle \text{For all }x\text{, is }x \leq 2x$ ? . . . No, not reflexive.

Symmetric: .$\displaystyle \text{If }x R\:\!y\text{, then }y R\:\!x.$
. . $\displaystyle \text{For all }x,y\text{, does }x \leq 2y \text{ imply }y \leq 2x$ ? . . . No, not symmtric.

Antisymmetric: .$\displaystyle \text{If }x R\:\!y\text{ and }y\:\!R\:\!x\text{, then: }x = y$
. . $\displaystyle \text{Does }x \leq 2y\text{ and }y \leq 2x \text{ imply }x = y$ ? . . . Yes, it is antisymmetric.

. . . . (It is true when $\displaystyle x = y = 0$)

Transitive: .If $\displaystyle x R\:\! y$ and $\displaystyle y\:\!R\:\! z$, then $\displaystyle x R z$
. . $\displaystyle \text{Does }x \leq 2y\text{ and }y \leq 2z\text{ imply }x \leq 2z$ ? . . . Yes, it is transitive.

3. Originally Posted by Storm20
Test the relation R for the properties of reflexivity, symmetry, antisymmetry and transitivity if R is on the set of positive integers, and is given by (x,y)eR if x<=2y
Originally Posted by Soroban
Reflexive: .$\displaystyle x R x$
. . $\displaystyle \text{For all }x\text{, is }x \leq 2x$ ? . . . No, not reflexive.
Antisymmetric: .$\displaystyle \text{If }x R\:\!y\text{ and }y\:\!R\:\!x\text{, then: }x = y$
. . $\displaystyle \text{Does }x \leq 2y\text{ and }y \leq 2x \text{ imply }x = y$ ? . . . Yes, it is antisymmetric.
. . . . (It is true when $\displaystyle x = y = 0$)
Please note that the underlying set is the set of positive integers.
Any positive integer is less that twice itself. Therefore, the relation is reflexive.

Now note that: $\displaystyle (1,2)\in R \text{ and } (2,1)\in R \text { but } 1 \not= 2$.
So can the relation be antisymmetric?

4. Thanx heaps for that guys, the one part of the assignment that I just COULDNT understand. Really appreciate it, thanx for explaining it too.

peace.

5. Hello, Plato!

Originally Posted by Plato

Please note that the underlying set is the set of positive integers.
Any positive integer is less that twice itself. Therefore, the relation is reflexive.

Now note that: $\displaystyle (1,2)\in R \text{ and } (2,1)\in R \text { but } 1 \not= 2$.
So can the relation be antisymmetric?

Absolutely right on both counts! . . . *blush*
Somehow, I misunderstood the role of that "2" . . . (slap head)