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