Thread: 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!

Nice question
I know some

1.a~b as " " (any equivalence relation works)
2.
3a~b as "a.b = a+b"
4 a~b as " "
5
6a~b as " " or "a|b" or " "
7a~b as " " (~ on Z for example)
8 a~b as

Ill try to think of more

Originally Posted by Dinkydoe

Ill try to think of more

Thanks for your help!
Hmm do you think that an example for 2. could be:
(x,x); x∈X (as the diagonal, the smallest reflexive relation on X )

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 = 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.

Hello, Dandelion!

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

There are 8 options:

. .


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.




Any relation involving "equality".
Examples: "is equal to", "is as old as", "is as tall as", etc.





Assume this involves a legal ceremony.

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

Symmetric: .

Not transtive: .
. . unless, of course, bigamy is legal on your planet.




Any relation involving "inequality".
Examples: "is older than", "is shorter than", etc.




" lives within one mile of "

lives within one mile of him/herself (reflexive).

live within one mile of each other (symmetric).

Not transitive: . does not imply
. . . and can be a mile apart.
. . . and can be a mile apart.
But and can be two miles apart.




is a brother of
Assume that this means: is male and has the same parents as

is a brother of himself (reflexive).

(transitive)

Not symmetric: . does not imply
. . can be a brother of , but can be a sister of

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 then answers #1.

The diagonal is reflexive but it is also transitive and symmetric. So it also satisfies #1.

The relation is reflexive but is neither symmetric nor transitive.

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

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!!