# Thread: Reflexive, symmetric, and transitive

1. ## Reflexive, symmetric, and transitive

I'm not sure I have these concepts down well enough. The following is a problem from my professor:

Let S = {1,2,3,4,5}. On P(S) define the relation $\rho$ as follows: A $\rho$B iff A $\cap$B = $\emptyset$. Is $\rho$ reflexive? Is $\rho$ symmetric? Is $\rho$ transitive?

Since there are elements in P(S) that do not equal $\emptyset$ when intersecting themself then $\rho$ is not reflexive.

If A $\cap$B is empty then B $\cap$A is empty so $\rho$ is symmetric.

{1,2} $\cap$ {3,4} is empty. And {3,4} $\cap$ {2,5} is empty. But {1,2} $\cap$ {2,5} = {2}. So $\rho$ is not transitive.

Is the above correct? If not please explain where I've gone wrong.

Happy KwanzHanukmas!

2. Originally Posted by oldguynewstudent
Let S = {1,2,3,4,5}. On P(S) define the relation $\rho$ as follows: A $\rho$B iff A $\cap$B = $\emptyset$. Is $\rho$ reflexive? Is $\rho$ symmetric? Is $\rho$ transitive?

Since there are elements in P(S) that do not equal $\emptyset$ when intersecting themself then $\rho$ is not reflexive.

If A $\cap$B is empty then B $\cap$A is empty so $\rho$ is symmetric.

{1,2} $\cap$ {3,4} is empty. And {3,4} $\cap$ {2,5} is empty. But {1,2} $\cap$ {2,5} = {2}. So $\rho$ is not transitive.
Yes, they are all correct.

3. Originally Posted by oldguynewstudent
I'm not sure I have these concepts down well enough. The following is a problem from my professor:

Let S = {1,2,3,4,5}. On P(S) define the relation $\rho$ as follows: A $\rho$B iff A $\cap$B = $\emptyset$. Is $\rho$ reflexive? Is $\rho$ symmetric? Is $\rho$ transitive?

Since there are elements in P(S) that do not equal $\emptyset$ when intersecting themself then $\rho$ is not reflexive.

If A $\cap$B is empty then B $\cap$A is empty so $\rho$ is symmetric.

{1,2} $\cap$ {3,4} is empty. And {3,4} $\cap$ {2,5} is empty. But {1,2} $\cap$ {2,5} = {2}. So $\rho$ is not transitive.

Is the above correct? If not please explain where I've gone wrong.

Happy KwanzHanukmas!
You left out "reflexive" but that is easy. Is $A\cap A$ empty?