Directed Graphs: Reflexive, Symmetric, Transitive

In summary, the student graphed a directed graph incorrectly and then tried to fix it. However, transitivity was not satisfied so the student had to fix it again.
  • #1
number0
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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.
 

Attachments

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  • #2
(1) and (4) are incorrect. The relation you graphed there are transitive.
 
  • #3
micromass said:
(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.
 

Attachments

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  • #4
(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.
 
  • #5
micromass said:
(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?
 
  • #6
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)...
 
  • #7
micromass said:
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?
 

Attachments

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  • #8
Yes, that looks fine!
 
  • #9
micromass said:
Yes, that looks fine!

Thank you very much.
 

1. What is a directed graph?

A directed graph is a type of graph in which the edges have a specific direction. This means that the relationship between two vertices is one-way, unlike in an undirected graph where the relationship is bidirectional.

2. What is a reflexive graph?

A reflexive graph is a directed graph in which every vertex has a self-loop, meaning an edge that connects the vertex to itself. This indicates that every vertex has a relationship with itself.

3. What does it mean for a graph to be symmetric?

A symmetric graph is one in which the direction of the edges can be reversed without changing the graph's structure. In other words, if there is an edge from vertex A to vertex B, there is also an edge from vertex B to vertex A.

4. What is a transitive graph?

A transitive graph is a directed graph in which there is a path from one vertex to another through a series of edges. This means that if there is an edge from vertex A to vertex B, and an edge from vertex B to vertex C, there is also an edge from vertex A to vertex C.

5. How are directed graphs used in real life?

Directed graphs have many practical applications, such as in social networks, transportation systems, and computer networks. They can be used to model relationships between individuals, the flow of traffic, or the connections between different devices in a network.

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