Re: Properties of Relations

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Originally Posted by

**MadSoulz** 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.

Re: Properties of Relations

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**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.

Re: Properties of Relations

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Originally Posted by

**MadSoulz** 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}.

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Re: Properties of Relations

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**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.

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Re: Properties of Relations

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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

Re: Properties of Relations

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Originally Posted by

**MadSoulz** $\displaystyle R_3$ 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.

Re: Properties of Relations

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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.

Re: Properties of Relations

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

Re: Properties of Relations

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Originally Posted by

**MadSoulz** $\displaystyle R_3$ 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?

Re: Properties of Relations

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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.

Re: Properties of Relations

Thanks to you all for your time.