I recently taught several of my classes about the three major properties of relations: reflexive, symmetric and transitive (which I'm going to assume you are familiar with...if not, you probably don't care much about this post anyway). In my math class experience, they were taught as:

Reflexive

for all x: x = x

Symmetric

for all x,y: if x = y then y = x

Transitive

for all x,y,z: if x = y and y = z then x = z

To which I had always responded with an internally muttered, "ummm, duh". But what they didn't tell me was that it's just not a big deal that 7=7. Of course it does. That's not what the reflexive property is really all about. Relations can be any kind of true/false comparison between any kinds of objects.

6 = 9 (false, compares numbers with "equals")

8 < 12 (true, compares numbers with "less than")

Angle X is complementary to angle Y (compares angles with "complementary)

p implies q (compares facts with "implies)

and more.

And the three properties might apply to all of them. So as the sets get more fun and the comparisons get more interesting, the kids actually get excited about deciding which properties apply. Here are the examples I use (and the properties they have):

**comparisons on numbers:**

equals (R, S, T)

greater than (T)

not-equal-to (S)

*[note: not-equal-to isn't transitive since the first and third numbers could be the same]*

divides-evenly (R, T)

**comparisons on angles:**

congruent (R, S, T)

complementary-to (S)

**comparisons on facts:**

implies (R, T)

*[note: the fact that it isn't symmetric is a good review of how the truth of a conditional says nothing about the truth of its converse]*

**comparisons on students in school:**

has-a-class-with (R, S)

has-English-class-with (R, S, T)

*[note: because you can only be taking one English class at a time]*

has-tied-the-shoes-of (R)

*[note: gets laughs]*

has-kissed-on-the-mouth (S)

*[note: i tell them this is PG-13 rated. gets big laughs and giggles and "eeew"s (when considering transitivity especially)]*

**comparisons on all people:**

sibling-of (S)

*[note: strictly defined as full biological or adoptive sibling...no half- or step-. not transitive for the same reason not-equal-to isn't]*

brother-of (none!)

I tend to think that a lot of people don't understand that math vocabulary and symbols are supposed to reflect the patterns of thought that go on in our heads. Math is not about an arbitrary set of rules to apply to bizarre symbols. The students think this is barely math, but I tell them it's very much math--it's just that they never realized what's really math.

And the next day, they all know *exactly* why every segment is congruent to itself: "Because congruence is reflexive!"

it seems to me that for discussions of RST, one should use only _binary_ relations: if our relation is called ~, for example, the usual "infix" notation x~y means that the ordered pair (x,y) is an element of the set of ordered pairs called ~ (recall that a relation, by definition,

_is_ a set of ordered pairs).

the definitions you've pointed at (using the world's most famous _equivalence_ relation) seem to depend on the infix notation.

how are we to speak of reflexivitiy, for example, if we consider a trinary relation? i don't get it.

Posted by: vlorbik | October 01, 2005 at 02:27 PM

hmmm...

I'm not sure I follow your objection. All of the relations in my list are intended to be taken as binary relations.

X has-a-class-with Y, where X and Y are students, means that (X,Y) is in the set of ordered pairs that includes all pairs of students that have a class together.

I don't think any of my relations are trinary. Perhaps I need to make some of the definitions more clear?

Posted by: Polymath | October 01, 2005 at 02:56 PM

my fault entirely; sorry.

i careless took your lines

congruent (R, S, T)

complementary-to (S)

as refering to a trinary

and unary relation respectively

(didn't notice that you'd

*clearly stated* that you were

providing "answers"; in the

unlikely event that anybody

else is still so confused,

R, S, and T have their "obvious"

meanings (reflexive, ...).

like that old lady used to say,

"never mind ..." ...

Posted by: vlorbik | October 01, 2005 at 10:15 PM

Not-equal-to is symmetric, but it's not reflexive.

Posted by: Xerxes1729 | October 09, 2005 at 12:58 PM

ah yes, thanks xerxes...the post now reflects the correction.

Posted by: Polymath | October 09, 2005 at 01:09 PM

Do you by any chance have the lesson plan you used to teach this lesson? I am a beginning teacher interview for a position and I have to teach this for my demo lesson. Any help would be much appreciated.

Posted by: Mrs. Scarpa | April 27, 2006 at 10:45 PM