# Thread: reflexive, symetric,transitive - examples?

1. ## reflexive, symetric,transitive - examples?

Hello,
I was wondering if someone could help me with this problem:

A relation on set can be reflexive (R), symmetric (S) or transitive(T) (..of course ) Depending on which of these ''basic characteristics'' it has, there are 8 options:
1. R, S, T
2.only R
3.only S
4.only T
5. R,S
6.R,T
7.S,T
8. neither R nor S nor T
I have to find an example for each option. I guess it's very easy, but somehow I have difficulties finding them, especially the ones that have only one characteristic. The first one I have of course : ) Any help is appreciated!

2. Nice question
I know some

1.a~b as " $a=b$" (any equivalence relation works)
2.
3a~b as "a.b = a+b"
4 a~b as " $a< b$"
5
6a~b as " $a\subset b$ " or "a|b" or " $\leq$"
7a~b as " $\max\left\{a,b\right\}< a+b$" (~ on Z for example)
8 a~b as $a+1 = b$

Ill try to think of more

3. Originally Posted by Dinkydoe
Ill try to think of more
Hmm do you think that an example for 2. could be:
(x,x); x∈X (as the diagonal, the smallest reflexive relation on X )

4. I'm not so sure. Then the relation is certainly reflexive, but probably symmetric and transitive too. "if" a~b then b~a since a=b in this case.

I hope you've had some topology:

Taking the standard topology on R: Let A~B <-->A\B is open
Reflexivity: A\A = $\emptyset$ is open.

Not symmetric: Let A = (0,1), B={2} then A\B = A is open. B\A= B is closed.
Not transitive: Let A = (0,1) and B=(2,3), C= (0,1/2) Then A\B=(0,1) is open, B\C=(2,3) is open. But A\C = [1/2,1) is not open.

There must be very simple examples, but I can't find one.

5. Hello, Dandelion!

A relation on set can be reflexive (R), symmetric (S) or transitive(T).

There are 8 options:

. . $\begin{array}{c| ccc}1&R&S&T \\2&R&-&- \\ 3.&-&S&- \\ 4.&-&-&T\\ 5.&R&S&- \\ 6.&R&-&T \\ 7.&-& S&T \\ 8.&-&-&-
\end{array}$

I have to find an example for each option.
I guess it's very easy. . .
no, it isn't!
This really stretches one's imagination.

$1.\;R,S,T$
Any relation involving "equality".
Examples: "is equal to", "is as old as", "is as tall as", etc.

$3.\:S\text{ only}$
$"X\text{ is married to }Y."$
Assume this involves a legal ceremony.

Not reflexive: . $a$ is not married to him/herself.

Symmetric: . $(a\circ b) \:\to\: (b\circ a)$

Not transtive: . $\bigg[(a\circ b) \wedge (b \circ c)\bigg] \;\rlap{\;\;/}\to\;(a\circ c)$
. .
unless, of course, bigamy is legal on your planet.

$4.\;T\text{ only}$
Any relation involving "inequality".
Examples: "is older than", "is shorter than", etc.

$5.\;R\text{ and }S\text{ only}$
" $X$ lives within one mile of $Y.$"

$a\circ a:\;\;a$ lives within one mile of him/herself (reflexive).

$\text{If } a\circ b\text{, then }b\circ a:\;\;a\text{ and }b$ live within one mile of each other (symmetric).

Not transitive: . $a\circ b \text{ and }b\circ c$ does not imply $a \circ c$
. . . $a$ and $b$ can be a mile apart.
. . . $b$ and $c$ can be a mile apart.
But $a$ and $c$ can be two miles apart.

$6.\;R\text{ and }T\text{ only}$
$"X$ is a brother of $Y."$
Assume that this means: $X$ is male and has the same parents as $Y.$

$a\circ a:\;\; a$ is a brother of himself (reflexive).

$\text{If }a\circ b\text{ and }b\circ c\text{, then }a\circ c\!:$ (transitive)

Not symmetric: . $a\circ b$ does not imply $b \circ a.$
. . $a$ can be a brother of $b$, but $b$ can be a sister of $a.$

$8.\;\sim\!R,\:\sim\!S,\:\sim\!T$
$"X\text{ is the father of }Y."$

6. Originally Posted by Dandelion
A relation on set can be reflexive (R), symmetric (S) or transitive(T) (..of course ) Depending on which of these ''basic characteristics'' it has, there are 8 options:
1. R, S, T
2.only R
3.only S
4.only T
5. R,S
6.R,T
7.S,T
8. neither R nor S nor T
While the replies are correct, there is an easier way.
Let $A=\{p,q,r\}$ then $A \times A$ answers #1.

The diagonal $\Delta_A=\{(p,p),(q,q),(r,r)\}$ is reflexive but it is also transitive and symmetric. So it also satisfies #1.

The relation $\Delta_A\cup\{(p,q),(q,r)\}$ is reflexive but is neither symmetric nor transitive.

Using this simple set we can continue to find examples for all eight.

7. Thank you all for your help and kindness!
I think I can definitely write the answer now. I might try to combine those different methods, although, I must admit, I didn't know we could use human relations as a mathematical example =)

Anyway, thanks once again and Happy New Year!!