# Reflexive, symmetric, and transitive

• Dec 5th 2009, 10:11 AM
oldguynewstudent
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 $\displaystyle \rho$ as follows: A$\displaystyle \rho$B iff A$\displaystyle \cap$B = $\displaystyle \emptyset$. Is $\displaystyle \rho$ reflexive? Is $\displaystyle \rho$ symmetric? Is $\displaystyle \rho$ transitive?

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

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

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

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

Happy KwanzHanukmas!
• Dec 5th 2009, 10:21 AM
Plato
Quote:

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

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

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

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

Yes, they are all correct.
• Dec 5th 2009, 11:08 AM
HallsofIvy
Quote:

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 $\displaystyle \rho$ as follows: A$\displaystyle \rho$B iff A$\displaystyle \cap$B = $\displaystyle \emptyset$. Is $\displaystyle \rho$ reflexive? Is $\displaystyle \rho$ symmetric? Is $\displaystyle \rho$ transitive?

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

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

{1,2} $\displaystyle \cap$ {3,4} is empty. And {3,4} $\displaystyle \cap$ {2,5} is empty. But {1,2} $\displaystyle \cap$ {2,5} = {2}. So $\displaystyle \rho$ 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 $\displaystyle A\cap A$ empty?