1. ## properties of relations

Which properties does each of the following relations satisfy among reflexive, symmetric, transitive, irreflexive, and antisymmetric.

R = {(a,b) l a squared = b squared} over the real numbers

R = {(x,y) l x divides y} over the positive integers

I thought the first one might be irreflexive, but I am not sure, and I don't know on the second one.

2. Originally Posted by patches
Which properties does each of the following relations satisfy among reflexive, symmetric, transitive, irreflexive, and antisymmetric.

R = {(a,b) l a squared = b squared} over the real numbers

R = {(x,y) l x divides y} over the positive integers

I thought the first one might be irreflexive, but I am not sure, and I don't know on the second one.
$\displaystyle \forall$ a,b,c
1) reflexive does(a,a) $\displaystyle \in R$?
2) symmetric, if (a,b) $\displaystyle \in R$, is (b,a) $\displaystyle \in R$?
3) transitive, if (a,b) $\displaystyle \in R$, is (b,a) $\displaystyle \in R$?
4) irreflexive, does (a,a) $\displaystyle \notin R$?
5) antisymmetric, if (a,b) $\displaystyle \in R$, and (b,a) $\displaystyle \in R$ then is a=b?

A)
R = {(a,b) l a squared = b squared} over the real numbers
"Equal" is a reflexive, symmetric, transitive relation
Try all the 5 conditions out for R = {(x,y) l x = y}

B)
R = {(x,y) l x divides y} over the positive integers
Note: I will use a|b for "a divides b"

Start checking:
1) a|a
2) Not symmetric , See 5)
3)if a|b and b|c then b = au and c = bv. Thus c = bv = (au)v = a(uv) and therefore a|c.
4) Since 1) holds, 4) cant
5) if a|b and b|a then $\displaystyle a = \pm b$.But since a,b $\displaystyle \in \mathbb{Z}^+$, a=b.