1. ## 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 $\displaystyle R=\left\{(1,1),(3,3),(1,3)\right\}$.

2. ## Re: 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 $\displaystyle \Delta_A=\left\{(1,1),(2,2),(3,3),(4,4)\right\}$ 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.

3. ## Re: Properties of Relations

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.

4. ## Re: Properties of Relations

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}.

5. ## Re: Properties of Relations

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.

6. ## Re: Properties of Relations

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

7. ## Re: Properties of Relations

$\displaystyle R_3$ 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.

8. ## Re: Properties of Relations

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.

9. ## Re: Properties of Relations

$\displaystyle R_3$ was the only relation that was not reflexive. Forgive my earlier failed explanation

10. ## Re: Properties of Relations

$\displaystyle R_3$ 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?

11. ## Re: Properties of Relations

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.

12. ## Re: Properties of Relations

Thanks to you all for your time.