Abstract Algebra Reflexive Properties

• Oct 29th 2006, 05:52 PM
henshilwood
Abstract Algebra Reflexive Properties
Need help understanding reflexive properties of Abstract algebra. If A = {1, 2, 3} what are its reflexive properties?
• Oct 29th 2006, 06:09 PM
ThePerfectHacker
Quote:

Originally Posted by henshilwood
Need help understanding reflexive properties of Abstract algebra. If A = {1, 2, 3} what are its reflexive properties?

Given a set $\displaystyle A$ and a relation $\displaystyle R$ on $\displaystyle A$ a reflexsive property is defined as if $\displaystyle yRx$ then, $\displaystyle xRy$. Now the meaning of $\displaystyle yRx$ is $\displaystyle (y,x)\in R$.

The reflexive property appears in two specific types of relations. An equivalence relation used a lot in abstract algebra and partial ordering relation a favorite in set theory.

I cannot answer thy question for thee has not defined a relation of $\displaystyle \{1,2,3\}$
• Oct 29th 2006, 06:16 PM
henshilwood
Quote:

Originally Posted by ThePerfectHacker
Given a set $\displaystyle A$ and a relation $\displaystyle R$ on $\displaystyle A$ a reflexsive property is defined as if $\displaystyle yRx$ then, $\displaystyle xRy$. Now the meaning of $\displaystyle yRx$ is $\displaystyle (y,x)\in R$.

The reflexive property appears in two specific types of relations. An equivalence relation used a lot in abstract algebra and partial ordering relation a favorite in set theory.

I cannot answer thy question for thee has not defined a relation of $\displaystyle \{1,2,3\}$

The specific question out of the book is "Let A be the set {1, 2, 3}. List the ordered pairs in a relation on A which is reflexive."
• Oct 29th 2006, 06:48 PM
ThePerfectHacker
Quote:

Originally Posted by henshilwood
The specific question out of the book is "Let A be the set {1, 2, 3}. List the ordered pairs in a relation on A which is reflexive."

Sorry, but the important thing is that you understand what I said. Relations are fundamental to understand algebra.

Okay the following set,
$\displaystyle \{ (1,2),(2,1),(1,3),(3,1),(2,3), (3,2)\}$
Is a relation on $\displaystyle \{1,2,3\}$.
• Oct 30th 2006, 02:56 AM
topsquark
Quote:

Originally Posted by ThePerfectHacker
Sorry, but the important thing is that you understand what I said. Relations are fundamental to understand algebra.

Okay the following set,
$\displaystyle \{ (1,2),(2,1),(1,3),(3,1),(2,3), (3,2)\}$
Is a relation on $\displaystyle \{1,2,3\}$.

We should note that the relation that TPH provided is, in a sense, "maximal" since it includes all the points in A. But we could also say that: $\displaystyle \{ (1,2), (2,1) \}$ is also a reflexive relation on A. (And similar other relations.)

-Dan
• Oct 30th 2006, 04:03 AM
ThePerfectHacker
Quote:

Originally Posted by topsquark
We should note that the relation that TPH provided is, in a sense, "maximal" since it includes all the points in A. But we could also say that: $\displaystyle \{ (1,2), (2,1) \}$ is also a reflexive relation on A. (And similar other relations.)

-Dan

Funny thing is I was working with the symmettic relation all of time and did not realize it. Hope I did not confuse the reader.
• Oct 30th 2006, 04:55 AM
Plato
The relation $\displaystyle \Delta _A = \left\{ {\left( {1,1} \right),\left( {2,2} \right),\left( {3,3} \right)} \right\}$ is called the diagonal relation. Any relation that contains the diagonal is a reflexive relation. Now there are 6 off-diagonal pairs in AxA. Thus, there are $\displaystyle 2^6$ subsets of the off-diagonal pairs. Unite each of those with the diagonal to get a reflexive relation on A. That is, there are $\displaystyle 2^6$ reflexive relations on A.
• Oct 30th 2006, 08:16 AM
ThePerfectHacker
Quote:

Originally Posted by Plato
The relation $\displaystyle \Delta _A = \left\{ {\left( {1,1} \right),\left( {2,2} \right),\left( {3,3} \right)} \right\}$ is called the diagonal relation. Any relation that contains the diagonal is a reflexive relation. Now there are 6 off-diagonal pairs in AxA. Thus, there are $\displaystyle 2^6$ subsets of the off-diagonal pairs. Unite each of those with the diagonal to get a reflexive relation on A. That is, there are $\displaystyle 2^6$ reflexive relations on A.

But the problem is the user was asking for a relation that is reflexsive but not symettric and transitive. The reason why I know is because I spoke with him on PM.
• Oct 30th 2006, 08:59 AM
Plato
Well, that is certainly not what he wrote.
I took him at his word: “Need help understanding reflexive properties ..”.