# Properties of Relations

• Nov 7th 2009, 05:15 AM
feliks0
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

(Hi)
• Nov 7th 2009, 05:29 AM
Plato
Your images are so poor I find it almost impossible to read them.
$R_1$ is not transitive: $(3,4) \in R_1 \wedge (4,1) \in R_1 \wedge (3,1) \notin R_1$.
• Nov 7th 2009, 07:03 AM
feliks0
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?
• Nov 7th 2009, 07:50 AM
Plato
What does it mean to be the complement of a relation?
What does it mean for a relation to be connected?
• Nov 7th 2009, 07:56 AM
feliks0
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?
• Nov 7th 2009, 08:04 AM
Plato
Quote:

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 $R_1~\&~(R_1)'$.
• Nov 7th 2009, 08:23 AM
feliks0
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>}
• Nov 7th 2009, 08:46 AM
Plato
Well according to that definition $(R_1)’$ is clearly connected.

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