How does relation R on A not satisfy these conditions?

Question

I think my math professor might have corrected my test questions wrong.

I got wrong on 3 questions that look right to me, they are about relationships on sets.

The questions:

Let A = {1, 2, 3, 4} and give an example of a relation R on A, so that the relation is

a) reflexive and symmetric but not transitive

b) symmetric and transitive but not reflexive

c) reflexive and transitive but not symmetric

![My Solutions][1]

What is wrong with my answers, I can't for the life of me see it.

Answer 1

a. The professor is right: that relation is transitive, seeing as $R\circ R=R$.

b. Your professor is right: the relation is not transitive since you have $1R2\land 2R1\land \neg 1R1$.

c. I see a red $\checkmark$. Doesn't that denote a good answer?

Answer 2

Your answer for (a) is in fact transitive. Do not fall into the trap of thinking that just because three different letters appear in most ways of writing the definition for transitivity that those letters must represent distinct elements.

You were asked to find an example for (a) which was not transitive. A better example would have been $\{(1,1),(2,2),(3,3),(4,4),(1,2),(2,1),(2,3),(3,2)\}$

For (b), again... you are misunderstanding transitivity. As before, do not fall into the trap of thinking the three different letters must represent distinct elements... they don't. You have for instance $(1,2)$ and $(2,1)$ in your relation however you are missing $(1,1)$ in your relation.

Recall... a relation is transitive iff for all choices of $\color{red}{x},\color{blue}{y},\color{green}{z}$ (not necessarily representing distinct elements) if you have $(\color{red}{x},\color{blue}{y})$ and $(\color{blue}{y},\color{green}{z})$ both in the relation then you must also have $(\color{red}{x},\color{green}{z})$.

Here, you had again $(\color{red}{1},\color{blue}{2})$ and $(\color{blue}{2},\color{green}{1})$ in the relation but you did not have $(\color{red}{1},\color{green}{1})$ in your relation, so your example was in fact not transitive.

A better answer for (b) would have been the empty relation which will in fact satisfy all of your desired properties. Alternatively, if the empty relation bothers you (which it shouldn't) another example which should be easier for a beginner to verify is $\{(1,1),(2,2),(3,3),(1,2),(1,3),(2,1),(2,3),(3,1),(3,2)\}$ over the set $\{1,2,3,4\}$.

For (c), this was correct and it appears your professor marked it as correct... note the downward and upward stroke of the pen here as compared to the incorrect ones who have only upward markings

For (d), this is in fact all three... reflexive, symmetric, and transitive. Again, this happens because the definitions for transitivity and symmetry do not specify distinct values for the variables involved. This is the usual "equality" relation which you should know is the prototypical example of an equivalence relation.

For (e) and (f), these are correct and were marked as such.