# Thread: Relations of sets (again)

1. ## Relations of sets (again)

This is my question :-

The relation T on the set {a,b,c,d} is given by the set of pairs

{(a,a), (b,b), (c,c), (a,c), (a,d), (b,d), (c,a), (d,a)}

Find each of
a)The reflexive closure of T
b)The symetric closure of T
c)The transitive closure of T

As I said with the previous question...

I apologize now for the fact I have not done any working out simply because I don't have a clue how to even start it! I'm trying to learn it now as I go along but don't have much time so thought I would look for help on here as well as the text book (which isn't making sense right now) so I have two points of view on the question. I'm stuck on two more questions which I will post on sperate threads too. Thanks for any help it's much appreciated.

2. Originally Posted by djmccabie
This is my question :-

The relation T on the set {a,b,c,d} is given by the set of pairs

{(a,a), (b,b), (c,c), (a,c), (a,d), (b,d), (c,a), (d,a)}

Find each of
a)The reflexive closure of T
b)The symetric closure of T
c)The transitive closure of T

As I said with the previous question...

I apologize now for the fact I have not done any working out simply because I don't have a clue how to even start it! I'm trying to learn it now as I go along but don't have much time so thought I would look for help on here as well as the text book (which isn't making sense right now) so I have two points of view on the question. I'm stuck on two more questions which I will post on sperate threads too. Thanks for any help it's much appreciated.
What does the closure of a property mean?

3. What does the closure of a property mean?
I don't see where the problem refers to the closure of a property. It talks about the reflexive closure of a relation $\displaystyle T$. It is the smallest relation that contains $\displaystyle T$ and is reflexive. Obviously, $\displaystyle T\cup\{(a,a),(b,b),(c,c),(d,d)\}$ is the reflexive closure of $\displaystyle T$.

4. Originally Posted by djmccabie
This is my question :-

The relation T on the set {a,b,c,d} is given by the set of pairs

{(a,a), (b,b), (c,c), (a,c), (a,d), (b,d), (c,a), (d,a)}

Find each of
a)The reflexive closure of T
b)The symetric closure of T
c)The transitive closure of T

As I said with the previous question...

I apologize now for the fact I have not done any working out simply because I don't have a clue how to even start it! I'm trying to learn it now as I go along but don't have much time so thought I would look for help on here as well as the text book (which isn't making sense right now) so I have two points of view on the question. I'm stuck on two more questions which I will post on sperate threads too. Thanks for any help it's much appreciated.
Take A = {a,b,c,d}

Let E denote the diagonal subset of AxA.

Let T^c denote the relation that's the converse of T.

Let T^n denote the composition of T with itself n times.

The reflexive closure is just T U E.

The symmetric closure is T U T^c.

The transitive closure is, in general, T U T^2 U ... .

In a fixed finite case, to get it, you don't have to go out any farther than, T U T^2 U ... U T^n, where n = |A|.
Here n=4 (pretty small), so you can just grind out its extension.