It seems that we fairly regularly get elementary questions of the form:
"Does a relation that is [A] and [B] have to be [C]?"
where [A], [B] and [C] are some permutation of "transitive", "symmetric" and "reflexive" or their negations.
For example, here's a recent one that made it onto the Hot Network Questions list.
Of course, the answer to all these questions is "no, here's a basic counterexample." In fact, a couple of years ago I posted an answer that provides counterexamples to all these questions. For convenience, let me briefly quote that answer here:
"On the three-element set $\{a, b, c\}$, the following relations are:
- Transitive, symmetric: $R_0 = \emptyset$
- Transitive, not symmetric: $R_1 = \{(a,b)\}$
- Not transitive, not symmetric: $R_2 = \{(a,b), (b,c)\}$
- Not transitive, symmetric: $R_3 = \{(a,b), (b,a), (b,c), (c,b)\}$
None of the relations above are reflexive, but they can all be turned into reflexive relations, without affecting their transitivity or symmetry, by adding $R^* = \{(a,a), (b,b), (c,c)\}$ to them."
I'm not saying that it's a perfect answer — one could surely do much more to try and build the reader's intuition for why these properties are independent — but it does show that a single answer (even a fairly short one) is sufficient to cover all these questions.
Unfortunately, the question that I originally posted that answer under is rather more narrowly scoped than the answer itself. Prompted by the recent Hot Network Question, I was think that I might want to post a more broader "canonical" question that did cover all these variations, and repost the answer (somewhat expanded, perhaps with some pictures to illustrate it, and probably as Community Wiki to invite improvements from others) under it.
That canonical question might, tentatively, look something like this:
"In school, we learned about symmetric, transitive and reflexive relations. Do any two of these properties (or their negations) imply the third (or its negation), or are they all independent?"
(There would, of course, also be a link to this meta question.)
That said, before going ahead with this, I figured I'd ask whether anyone else thinks this is a good idea. Would a single canonical question like this be a useful resource, or should we just maintain separate questions with separate answers for each of the (3 × 2³ = 24 or so) specific permutations of this question?
