1. ## Properties of Relations

Hello,

I uploaded a question concerning properties of relations which I solved.
When I wanted to double-prove my answers, I found some conflicts.
Please have a look at the original question as well as the solution offered by the book.

1) Concerning the relation R1:

I find R1 and its inverse to be transitive relations whereas in the solution it is written that those are non transitive relations. I wonder where one of the conditions of a transitive relation is violated in R1.

2) Concerning the complement relation of R1:

Another conflict is raised. I find the relation to be connected whereas it is stated that the relation is non connected.

For your convenience the original relation is presented in the question.

Felix

2. Your images are so poor I find it almost impossible to read them.
$\displaystyle R_1$ is not transitive: $\displaystyle (3,4) \in R_1 \wedge (4,1) \in R_1 \wedge (3,1) \notin R_1$.

3. Thank you very much! That solved the first conflict. Now I'm left with the connected/non connected issue.

Anyone has an idea why the complement of R1 is non connected, whereas I find it connected?

4. What does it mean to be the complement of a relation?
What does it mean for a relation to be connected?

5. Well, according to the definitions in my book, if R is a relation then R' is the complement of the relation, that is a relation that includes the set of ordered pairs which do not appear in the original R relation.

And connectedness is one of the properties of a relation that are discussed in the book. Shall I quote the exact definitions?

6. Originally Posted by feliks0
And connectedness is one of the properties of a relation that are discussed in the book. Shall I quote the exact definitions?
Yes quote the definition of connectedness.
I still find the images impossible to read.
So if you want help, type out both $\displaystyle R_1~\&~(R_1)'$.

7. The definition of connectedness-

A relation R in A is connected iff for every two distinct elements x and y in A, <x,y> is a member of R or <y,x> is a member of R (or both).

Examples:

A = {1,2,3}

{<1,2>, <3,1>, <3,2>}

{<1,1>, <2,3>, <1,2>, <3,1>, <2,2>}

nonconnected-

{<1,2>, <2,3>}
{<1,3>, <3,1>, <2,2>, <3,2>}

Now back to the relation R1'-

{<1,2>, <1,3>, <1,4>, <2,3>,<2,4>, <3,1>, <3,2>,<4,2>, <4,3>}

8. Well according to that definition $\displaystyle (R_1)’$ is clearly connected.

But $\displaystyle R_1$ is not connected for $\displaystyle (2,4)\notin R_1~\&~(4,2)\notin R_1$.