Hi. I have this problem.

Let be a binary relation on a set , which is not necessarily transitive.

Define a binary relation on as follows: for any , where is an integer (which relies on and ), such that and for all . is called the transitive closure of .

Prove that is a transitive relation on .

This is what I did... but not sure if it really make sense though.

Suppose and are in .

Show that is in .

By definition of ,

is in for some

is in for some

Then is in which is contained in . Hence must be transitive. i.e. is a transitive relation on .

What I'm not sure what I have done is choosing an artribrary number in the sequence between and is correct, it is where I chose . Also shall I use and instead of and in my proof? Or should i just stick with and ?

Thanks heaps for your help.