# Thread: Relations and Their Properties

1. ## Relations and Their Properties

I am confused on what to do with this problem from my textbook because it does not give me sets. For example: A = {1, 2, 3} and B = {1, 2, 3, 4} and R1 = {(1,1), (2,2), (3,3), (4,4)} and R2 = {(1,1), (1,2), (1,3, (1,4)}, then I can do the unions and intersection, etc but I do not have that options here.

What I have is this as my problem:

R1 = {(a, b) R^2 | a > b}, the "greater than" relation
R2 = {(a, b) R^2 | a > b}, the "greater than or equal to" relation
R3 = {(a, b) R^2 | a < b}, the "less than" relation
R4 = {(a, b) R^2 | a < b}, the "less than or equal to" relation
R5 = {(a, b) R^2 | a = b}, the "equal to" relation
R6 = {(a, b) R^2 | a b}, the "equal to" relation

Find:
a.) R1 U R3
b.) R1 U R5
c.) R2 R4
d.) R3 R5
e.) R1 - R2
f.) R2 - R1
g.) R1 R3
h.) R2 R4

The R1, R2, R3... the numbers are subscripts.

Also If I have this, what do I do if i have this white circle in between them, not up high. For example:
a.) R1 R1

I have 5 problems similar to this one.

2. I will be glad to give you the answers to three of these as representatives of the others.
BUT, I will not give them with explanations. I expect you to puzzle out the reasons.
Then you should respond with the solutions to the others.
$\begin{gathered}
a)~\mathcal{R}_1 \cup \mathcal {R}_2 = \Re ^2 \hfill \\
d)~\mathcal {R}_3 \cap \mathcal {R}_5 = \emptyset \hfill \\
h)~\mathcal{R}_2 \oplus \mathcal {R}_4 = \mathcal {R}_6 \hfill \\
\end{gathered}$

3. Originally Posted by Plato
I will be glad to give you the answers to three of these as representatives of the others.
BUT, I will not give them with explanations. I expect you to puzzle out the reasons.
Then you should respond with the solutions to the others.
$\begin{gathered}
a)~\mathcal{R}_1 \cup \mathcal {R}_2 = \Re ^2 \hfill \\
d)~\mathcal {R}_3 \cap \mathcal {R}_5 = \emptyset \hfill \\
h)~\mathcal{R}_2 \oplus \mathcal {R}_4 = \mathcal {R}_6 \hfill \\
\end{gathered}$
Plato, can you show me how to do part a so it gives me an idea of what to do this way I can try to solve the others and run it by you to see if its correct? I do not understand how you got R^2

4. Originally Posted by Grillakis
Plato, can you show me how to do part a so it gives me an idea of what to do this way I can try to solve the others and run it by you to see if its correct? I do not understand how you got R^2
Do you know what the Trichotomy Property for real numbers?
If not, it is sad to say that you are under prepared to do this exercise.
If each of $a\text{ and } b$ is a real number then exactly one of these is true $a>b,~a=b,~ a.

5. Originally Posted by Plato
Do you know what the Trichotomy Property for real numbers?
If not, it is sad to say that you are under prepared to do this exercise.
If each of $a\text{ and } b$ is a real number then exactly one of these is true $a>b,~a=b,~ a.
We have not done any Trichotomy property. We did Transitive, Symmetry, Antisymmetry, Reflective, and Irreflective. Thats what we covered and this problem is part of the exercise in this chapter.

Well I will see what I can do with the other problem and run it by you to see if I am doing this correct.

6. Originally Posted by Grillakis

Find:
a.) R1 U R3
b.) R1 U R5
c.) R2 R4
d.) R3 R5
e.) R1 - R2
f.) R2 - R1
g.) R1 R3
h.) R2 R4
Plato, I got this:

b.) R2
c.) R5
e.) empty set symbol
f.) R5
g.) R6