# Properties of Relations

• Nov 16th 2013, 09:38 AM
Properties of Relations
course: Foundations of Higher Math

Let $\displaystyle A=\left\{1,2,3,4\right\}$. Give an example of a relation on A that is reflexive and transitive, but not symmetric.

My question is, does my relation have to include all of the pairs $\displaystyle \left\{(1,1),(2,2),(3,3),(4,4)\right\}$ to be reflexive?

For example, say my relation is $\displaystyle R=\left\{(1,1),(3,3),(1,3)\right\}$.
• Nov 16th 2013, 09:45 AM
Plato
Re: Properties of Relations
Quote:

course: Foundations of Higher Math

Let $\displaystyle A=\left\{1,2,3,4\right\}$. Give an example of a relation on A that is reflexive and transitive, but not symmetric.

My question is, does my relation have to include all of the pairs $\displaystyle \Delta_A=\left\{(1,1),(2,2),(3,3),(4,4)\right\}$ to be reflexive?

For example, say my relation is $\displaystyle R=\left\{(1,1),(3,3),(1,3)\right\}$.

Any reflexive relation on $\displaystyle A$ must contain $\displaystyle \Delta_A$ as a subset.
• Nov 16th 2013, 10:55 AM
Re: Properties of Relations
Quote:

Originally Posted by Plato
Any reflexive relation on $\displaystyle A$ must contain $\displaystyle \Delta_A$ as a subset.

Perhaps my professor made a mistake in class.

We had $\displaystyle S=\left\{1,2,3\right\}$ and said that
$\displaystyle R= \left\{(1,1),(2,2)\right\}$ was a reflexive relation.
• Nov 16th 2013, 11:43 AM
HallsofIvy
Re: Properties of Relations
Quote:

Perhaps my professor made a mistake in class.

We had $\displaystyle S=\left\{1,2,3\right\}$ and said that
$\displaystyle R= \left\{(1,1),(2,2)\right\}$ was a reflexive relation.

Are you sure you heard correctly? R is a reflexive relation on {1, 2} but not on {1, 2, 3}.
• Nov 16th 2013, 12:00 PM
Re: Properties of Relations
Quote:

Originally Posted by HallsofIvy
Are you sure you heard correctly? R is a reflexive relation on {1, 2} but not on {1, 2, 3}.

Yes, he wrote the following table on the board then filled it in himself.

$\displaystyle R_3$ was the only one that had an 'X' for reflexive.
• Nov 16th 2013, 12:11 PM
Re: Properties of Relations
Quote:

Originally Posted by HallsofIvy
Are you sure you heard correctly? R is a reflexive relation on {1, 2} but not on {1, 2, 3}.

Here's the entire Powerpoint slide
• Nov 16th 2013, 01:33 PM
emakarov
Re: Properties of Relations
Quote:

$\displaystyle R_3$ was the only one that had an 'X' for reflexive.

$\displaystyle R_3$ is not even close to being reflexive: it does not contain a single pair from the diagonal.
• Nov 16th 2013, 01:47 PM
Re: Properties of Relations
Quote:

Originally Posted by emakarov
$\displaystyle R_3$ is not even close to being reflexive: it does not contain a single pair from the diagonal.

Sorry

EDIT: $\displaystyle R_3$ was the only one that had an 'X' for *not* reflexive.
• Nov 16th 2013, 01:47 PM
Re: Properties of Relations
$\displaystyle R_3$ was the only relation that was not reflexive. Forgive my earlier failed explanation
• Nov 16th 2013, 01:59 PM
Plato
Re: Properties of Relations
Quote:

$\displaystyle R_3$ was the only one that had an 'X' for *not* reflexive.

Actually I suspected that was the case

$\displaystyle R_3$ is the only one in the list that is irreflexive: i.e. $\displaystyle \Delta_S\cap R_3=\emptyset~.$

I wonder if that is what your instructor meant?
• Nov 16th 2013, 02:07 PM
Re: Properties of Relations
Quote:

Originally Posted by Plato
Actually I suspected that was the case

$\displaystyle R_3$ is the only one in the list that is irreflexive: i.e. $\displaystyle \Delta_S\cap R_3=\emptyset~.$

I wonder if that is what your instructor meant?

Perhaps. I'll have to ask him on Monday. He doesn't answer emails on weekends.
• Nov 16th 2013, 02:12 PM