# Relation

• Jan 21st 2009, 07:53 PM
javax
Relation
Hello. I need help on determining this relation.
Determine the relation $\displaystyle \rho$ defined with $\displaystyle x\rho y \Leftrightarrow (\exists z)(y=zx)$ for $\displaystyle Z$ numbers
• Jan 21st 2009, 08:06 PM
Jhevon
Quote:

Originally Posted by javax
Hello. I need help on determining this relation.
Determine the relation $\displaystyle \rho$ defined with $\displaystyle x\rho y \Leftrightarrow (\exists z)(y=zx)$ for $\displaystyle Z$ numbers

what do you mean "determine the relation"? isn't it already determined. it is given.

note that, geometrically speaking, the relation describes the set of all straight lines that pass through the origin with integer slope (i suppose you mean $\displaystyle \mathbb{Z}$ when you wrote $\displaystyle Z$).
• Jan 21st 2009, 08:11 PM
javax
Quote:

Originally Posted by Jhevon
what do you mean "determine the relation"? isn't it already determined. it is given.

note that, geometrically speaking, the relation describes the set of all straight lines that pass through the origin with integer slope (i suppose you mean $\displaystyle \mathbb{Z}$ when you wrote $\displaystyle Z$).

Sorry for my english, I haven't had the chance to see these terms in english, I guess. Yes I mean $\displaystyle \mathbb{Z}$
I mean determine if it is equivalence relation or order relation. That's what it is required!

Thanks
• Jan 21st 2009, 08:42 PM
Jhevon
Quote:

Originally Posted by javax
Sorry for my english, I haven't had the chance to see these terms in english, I guess. Yes I mean $\displaystyle \mathbb{Z}$
I mean determine if it is equivalence relation or order relation. That's what it is required!

Thanks

it is not an equivalence relation. it is an (partial) order relation.

can you see why?
• Jan 22nd 2009, 08:29 AM
javax
Quote:

Originally Posted by Jhevon
it is not an equivalence relation. it is an (partial) order relation.

can you see why?

Nope, I need that prove. C'mon help me (Doh)
• Jan 22nd 2009, 11:14 AM
Jhevon
Quote:

Originally Posted by javax
Nope, I need that prove. C'mon help me (Doh)

here's how to start, what ae the definitions of "equivalence relation" and "(partial) order relations"? how do we know a relation is one of these or not?
• Jan 22nd 2009, 01:47 PM
javax
Quote:

Originally Posted by Jhevon

here's how to start, what ae the definitions of "equivalence relation" and "(partial) order relations"? how do we know a relation is one of these or not?

Here we go:
1. (r) $\displaystyle (\forall a\in A) a\rho a$
2. (s) $\displaystyle (\forall a,b \in A) a\rho b \Rightarrow b\rho a$
3. (t) $\displaystyle (\forall a,b,c \in A) a\rho b \wedge b\rho c \Rightarrow a\rho c$
4. (a)$\displaystyle (\forall a,b\in A)a\rho b \wedge b\rho a \Rightarrow a = b$

The relation that fulfills conditions 1, 2 & 3 is called equivalence relation. Relation that fulfills 1, 3 & 4 is called a partial order relation. That's what I know :)
• Jan 22nd 2009, 01:51 PM
Jhevon
Quote:

Originally Posted by javax
Here we go:
1. (r) $\displaystyle (\forall a\in A) a\rho a$
2. (s) $\displaystyle (\forall a,b \in A) a\rho b \Rightarrow b\rho a$
3. (t) $\displaystyle (\forall a,b,c \in A) a\rho b \wedge b\rho c \Rightarrow a\rho c$
4. (a)$\displaystyle (\forall a,b\in A)a\rho b \wedge b\rho a \Rightarrow a = b$

The relation that fulfills conditions 1, 2 & 3 is called equivalence relation. Relation that fulfills 1, 2 & 4 is called a partial order relation. That's what I know :)

actually, 1, 3 & 4 makes a partial order relation.

ok, so check if it is an equivalence relation:

(1) does x relate to itself? can you find and integer such that x = zx?
(2) do we have symmetry. if we can find and integer z so that y = zx, will we always be able to find an integer n so that x = ny?
(3) do we have transitivity? if we have integers n and m so that a = mb and b = nc, can we find an integer k so that a = kc?
• Jan 22nd 2009, 02:44 PM
javax
Quote:

Originally Posted by Jhevon
actually, 1, 3 & 4 makes a partial order relation.

ok, so check if it is an equivalence relation:

(1) does x relate to itself? can you find and integer such that x = zx?
(2) do we have symmetry. if we can find and integer z so that y = zx, will we always be able to find an integer n so that x = ny?
(3) do we have transitivity? if we have integers n and m so that a = mb and b = nc, can we find an integer k so that a = kc?

After exploring my notebook, here's my attempt:

(1) (r)
$\displaystyle x\rho x \Leftrightarrow \exists k=1$ such that $\displaystyle x=kx \Rightarrow x=x$
it fills the reflexive condition

(2) (s) $\displaystyle x\rho y (\exists z)(y=zx)$, $\displaystyle \hspace{2cm}y\rho x (\exists u)(x=uy)$
if we multiply these we have $\displaystyle (\exists z)(\exists u) yx=z x u y$ For $\displaystyle z=u=1$ we have $\displaystyle yx=xy$ (Does this mean that we have a symmetry?

(3) (t)$\displaystyle \exists z(y=zx)\wedge \exists u(t=uy)$. I multiplied again
$\displaystyle (\exists u)(\exists z) t=(uz)x.$
sub. $\displaystyle uz=p \Rightarrow t=px.$

What do you think?
• Jan 22nd 2009, 02:53 PM
Jhevon
Quote:

Originally Posted by javax
After exploring my notebook, here's my attempt:

(1) (r)
$\displaystyle x\rho x \Leftrightarrow \exists k=1$ such that $\displaystyle x=kx \Rightarrow x=x$
it fills the reflexive condition

brilliant!

Quote:

(2) (s) $\displaystyle x\rho y (\exists z)(y=zx)$, $\displaystyle \hspace{2cm}y\rho x (\exists u)(x=uy)$
if we multiply these we have $\displaystyle (\exists z)(\exists u) yx=z x u y$ For $\displaystyle z=u=1$ we have $\displaystyle yx=xy$ (Does this mean that we have a symmetry?
think of an example.

take y = 2 and x = 1. clearly y relates to x, since y = zx (where z = 2).

but does x relate to y? can we find an integer k so that x = ky, that is, 1 = k*2?

Quote:

(3) (t)$\displaystyle \exists z(y=zx)\wedge \exists u(t=uy)$. I multiplied again
$\displaystyle (\exists u)(\exists z) t=(uz)x.$
sub. $\displaystyle uz=p \Rightarrow t=px.$

What do you think?
yup, that's nice. we do have transitivity
• Jan 22nd 2009, 03:07 PM
javax
Quote:

Originally Posted by Jhevon

think of an example.

take y = 2 and x = 1. clearly y relates to x, since y = zx (where z = 2).

but does x relate to y? can we find an integer k so that x = ky

y=zx, x=ky
Ok like you said if we have y=x=z=k=1, we have a symmetry don't we?
• Jan 22nd 2009, 03:10 PM
Jhevon
Quote:

Originally Posted by javax
y=zx, x=ky
Ok like you said if we have y=x=z=k=1, we have a symmetry don't we?

i gave you an example where it didn't work, didn't i? look back at your definitions, it must work for ALL x and y, not just specific ones
• Jan 22nd 2009, 03:12 PM
javax
Quote:

Originally Posted by Jhevon
i gave you an example where it didn't work, didn't i? look back at your definitions, it must work for ALL x and y, not just specific ones

ohhhh cool(Rock)
ok so if it is not symmetric does that mean that it is antisymmetric?
• Jan 22nd 2009, 03:18 PM
Jhevon
Quote:

Originally Posted by javax
ohhhh cool(Rock)
ok so if it is not symmetric does that mean that it is antisymmetric?

no, anti-symmetric is defined by the fourth definition you have. after reading a few math books you should start to see how mathematicians define things. if anti-symmetric means "not symmetric" that is exactly how they would define it. it shouldn't be hard to come up with an example of a relation that is both not symmetric and not anti-symmetric. see here

you have to check if it is anti-symmetric. if it is, we have an order relation (since you already showed we have reflexivity and transitivity)