1. ## Examples of Relations...

Hello, I am stuck on a problem. The problem is:

Let A = {1, 2, 3}. Give an example of a relation R on A that is
a. Transitive, reflexive, but not antisymmetric
b. Antisymmetric, reflexive, but not transitive
c. Antisymmetric, transitive, but not reflexive

Ok, I know what each means,

Transitive: if (a,b) and (b,c) are in R, then (a,c) is in R.
Reflexive: if a is in A, then (a,a) is in R
Antisymmetric: For all a,b in A, if a =/= b and (a,b) are in R, then (b,a) is not in R.

That being said, I am having difficulty thinking of a relation to satisfy the conditions. Is there a good procedure to try for this type of problem? If anyone can give me a hint or shove in the right direction, that would be helpful. Thank you! I appreciate it.

Edit, I have given this some thought and have come up with this:

a. $\displaystyle R = {(a,b) \in Z X Z }$
b. $\displaystyle R = {(a,b) \in Z X Z, a = b}$
c. $\displaystyle R = {(a,b) \in Z X Z, a < b}$

2. Originally Posted by Alterah
Hello, I am stuck on a problem. The problem is:

Let A = {1, 2, 3}. Give an example of a relation R on A that is
a. Transitive, reflexive, but not antisymmetric
b. Antisymmetric, reflexive, but not transitive
c. Antisymmetric, transitive, but not reflexive

Ok, I know what each means,

Transitive: if (a,b) and (b,c) are in R, then (a,c) is in R.
Reflexive: if a is in A, then (a,a) is in R
Antisymmetric: For all a,b in A, if a =/= b and (a,b) are in R, then (b,a) is not in R.

That being said, I am having difficulty thinking of a relation to satisfy the conditions. Is there a good procedure to try for this type of problem? If anyone can give me a hint or shove in the right direction, that would be helpful. Thank you! I appreciate it.

Edit, I have given this some thought and have come up with this:

a. $\displaystyle R = {(a,b) \in Z X Z }$
b. $\displaystyle R = {(a,b) \in Z X Z, a = b}$
c. $\displaystyle R = {(a,b) \in Z X Z, a < b}$
What you've come up with looks a bit strange to me.

Let's just consider a., and leave b. and c. to someone else.

Think about R = {<1,1>, <2,2>, <3,3>, <1,2>, <2,3>, <2,1>, <1,3>}.
Convince yourself that R is indeed a preorder that does not have the additional property of antisymmetry.

In the finite case, there are effective methods for obtaining such results.
Establishing that might, in itself, be interestng.