Can someone please help me on this reflexive, transitive, and symmetric problem?

Question

Let $A = \{1,2,3\}$. Write the ordered pairs for a binary relation R on A which is reflexive and symmetric, but not transitive.

From my understanding is when listing if we want reflexive we need every example. However, for symmetric and reflexive providing ONE example is enough to show it. So if I wanted it to be transitive I can just do $\{(2,3),(3,1)$ and $(2,1)\}$. HOWEVER, when looking at solutions I'm extremely confused.

The solution is $\{(1,1),(2,2),(3,3)(1,2),(2,1),(2,3),(3,2)\}$ I get it's reflexive because (a,a) is always there. and I get it's symmetric because $(1,2),(2,1)$. Our prof says that $(1,2)$ and $(2,3)$ is in the list, but $(1,3)$ is not therefore it's not transitive.BUT how doesn't $(1,2),(2,1),(1,1)$ make it transitive. Isn't this one example enough to show its transitive???

Answer 1

Just think of the ordered pair $(a,b)$ as saying $a \sim b$. Because we have $(a,a)$ in the set for $a \in \{1,2,3\}$, we know that the relation is reflexive, after all, $(a,a)$ means $a \sim a$.

We know that the relation is symmetric, because for every element $(a,b)$ in the set, we have $(b,a)$ also in the set. That is, we have both $a \sim b$ (from $(a,b)$) and $b \sim a$ (from $(b,a)$).

We not transitive? Well, notice the set has $(1,2)$ so that $1 \sim 2$ and we have $(2,3)$ so that $2 \sim 3$. By transitivity, we should have $(1,3)$ in the set so that $1 \sim 3$, but notice it is not in the set! Therefore, the given $\sim$ is not transitive, and therefore not an equivalence relation.

Remember, it is not enough that you find an example of something transitive in the set, it has to be true for all of them! Meaning, every time you have $a \sim b$ and $b \sim c$ for some $a,b,c$, you then must have $a \sim c$ by transitivity. The example of $1 \sim 2$ and $2 \sim 3$ but $1 \not\sim 3$ is enough to show that the relation is not transitive.

So to show something is transitive, you need to show it for all of the possible applicable pairs. But to show something is not transitive, you only need to find a single example where it fails.

There may be an easier example. We will draw a 'graph' using the set $\{1,2,3\}$. We will say $a \sim b$ if either $a=b$ or if there is an edge connecting $a$ and $b$. Then draw the graph $$ 1 - 2 - 3 $$ Notice $1 \sim 1$, $2 \sim 2$, and $3 \sim 3$. We also have $1 \sim 2$ and $2 \sim 1$ because there is a 'segment' connecting them. Similarly, we have $2 \sim 3$ and $3 \sim 2$. But despite $1 \sim 2$ and $2 \sim 3$, we do not have $1 \sim 3$ because there is no direct 'segment' connecting $1$ and $3$. [There is a path through 2, but we dont want paths, we want direct paths.]

Edit. Note instead of the odd $a \sim b$ if $a=b$ or.... criterion, you could just draw a loop at each of the numbers $1,2,3$ connecting them with themselves....but TeXing that would be an irritation, so I opted for the former.

Answer 2

To be reflexive you need $(1,1),(2,2),(3,3)$ to be included. To be symetric we don't need to put anything in but if you put in $(a,b)$ you need $(b,a)$. The not be transitive we need a $(a,b), (b,c)$ but not $(a,c)$.

So put in $(1,2)$ and $(2,3)$ but not $(1,3)$. As we put in $(1,2)$ we need $(2,1)$ and as we put in $(1,3)$ we need $(3,1)$>

So $\{(1,1),(2,2),(3,3), (1,2),(2,1), (2,3),(3,2)\}$ will do. (note: the requires $7$ of the $9$ pairs)

Answer 3

BUT how doesn't (1,2),(2,1),(1,1) make it transitive. Isn't this one example enough to show its transitive???

You've misunderstood an important part of the definition. A relation is transitive if for every a, b, and c with (a,b) and (b,c) in the relation, (a,c) is also in the relation.

This is not the same as "There exist a, b, and c such that (a,b), (b,c), and (a,c) are in the relation."

You need to be more precise in your use of mathematical English and particularly in the use of quantifiers like "there exists" and "for all".

Answer 4

Isn't this one example enough to show its transitive???

No.

The statements describing the properties of Reflexivity, Symmetry, and Transitivity, are universally quantified.$${\text{Reflexivity:}~\forall x{\in}A~.x\operatorname R x\\\text{Symmetry:}~\forall x{\in}A~\forall y{\in}A~.(x\operatorname R y\to y\operatorname R x)\\\text{Transitivity:}~\forall x{\in}A~\forall y{\in}A~\forall z{\in}A~.(x\operatorname R y\wedge y\operatorname R z\to x\operatorname R z)}$$ As such, they are each held to be true exactly when the predicate is satisfied by every occurrence of the bound variables, and so they are each falsified by the existence of some counterexample.

$${\text{Non-Reflexive:}~\exists x{\in}A~.\neg x\operatorname R x\\\text{Non-Symmetry:}~\exists x{\in}A~\exists y{\in}A~.(x\operatorname R y\wedge\neg y\operatorname R x)\\\text{Non-Transitive:}~\exists x{\in}A~\exists y{\in}A~\exists z{\in}A~.(x\operatorname R y\wedge y\operatorname R z\wedge\neg x\operatorname R z)}$$