Directed Graphs: Reflexive, Symmetric, Transitive

[number0]

Homework Statement

Hello, I want to make sure that I graphed the directed graphs in my homework correctly. The problems and my work is located in the attachment. I also uploaded the directed graphs onto this link: http://img857.imageshack.us/f/83289329.png/"

Homework Equations

None

The Attempt at a Solution

In attachment + link.

Thanks.

 

[micromass]

(1) and (4) are incorrect. The relation you graphed there are transitive.

 

[number0]

(1) and (4) are incorrect. The relation you graphed there are transitive.

I do not understand why (4) is incorrect. But, I did changed somethings.

 

[micromass]

(4) is correct now. But (1) is still incorrect, since (1) is still transitive and it is not symmetric...

Transitive means: if you have an arrow from a to b, and an arrow from b to c, then you must have an arrow from a to c.

 

[number0]

(4) is correct now. But (1) is still incorrect, since (1) is still transitive and it is not symmetric...

Transitive means: if you have an arrow from a to b, and an arrow from b to c, then you must have an arrow from a to c.

Woops... silly of me. I misread the question. Still I do not understand why the first time I did (1) and (4) I got it wrong.

(1) Is it because I excluded c from a to b and as a result, a to b is transitive?

(4) Is it because I gave them no relationships so a, b, c is nothing?

 

[micromass]

Well, transitive says that IF there is a relation between a and b, and b and c, THEN there is a relation between a and c.

But in your (4), there is no relation between a and b, and b and c, thus transitivity is satisfied. You only need to check transitivity whenever there is a relation, if there is no relation then it is trivially satisfied. Thesame with your (1)...

 

[number0]

Well, transitive says that IF there is a relation between a and b, and b and c, THEN there is a relation between a and c.

But in your (4), there is no relation between a and b, and b and c, thus transitivity is satisfied. You only need to check transitivity whenever there is a relation, if there is no relation then it is trivially satisfied. Thesame with your (1)...

I see, then for (1), it should be something like this?

 

[micromass]

Yes, that looks fine!

 

[number0]

Yes, that looks fine!

Thank you very much.