Examples of relations: reflexive but not transtive; transtitive but not symmetric; symmetric but not reflexive

Question

Find example of a set $ S $ and three relations $R_1, R_2 ,R_3$ on it such that

$R_1$ is reflexive but not transitive,

$R_2$ is transitive but not symmetric and

$R_3$ is symmetric but not reflexive

I do not know how to start.

Can any one help, please?

Answer 1

Hint. Solutions to all the problems can be found by assuming that $S$ has fewer than (or equal to) $3$ elements.

A relation on a finite set $S$ can be visualized as a directed graph. Try drawing a few of these, especially with three vertexes.

Then try working out, under this point of view, what the words 'reflexive', 'symmetric' and 'transitive' correspond to.

In the end, you'll find the problem very easy to solve.

Answer 2

Let's start with the first part of the question. For simplicity, we will use a small set to work with, say {a, b, c}.

First, the definitions. A binary relation(we'll call it R) is $reflexive$ if (x, x) $\in$ R. And a relation is $transitive$ if (x , y) $\in$ R and (y, z) $\in$ R implies that (x, z) $\in$ R.

So an example of a reflexive relation that is not transitive on {a,b,c} would be the following relation:

{(a,a),(b,b),(c,c), (a,b), (b,c)}

Note that every element is in relation to itself, so it is reflexive. However, it is not transitive, because although a is in relation with b and b is in relation to c, a is not in relation c.

Do you think you can answer the other two parts of the question?

Answer 3

Following is the portion from Jim Hefferon's book on Linear Algebra

1. reflexivity: any object is related to itself;
2. symmetry: if a is related to b then b is related to a;
3. transitivity: if a is related to b and b is related to c then a is related to c.

In your problem the "object" is individual elements of set $S$

Examples of relation $R$ are $<, \leq, >, \geq, = $

Now make up a small set $S$, use different $R$ between the elements of $S$ for the answer.