00:01
In this problem, we have been asked to prove that a relation r is transitive if and only if r composition r is a subset of r.
00:09
So first of all, let us assume that r is transitive.
00:20
Then, let us consider xz to belong to r composition r.
00:28
Since xz belongs to r composition r, therefore there will exist y in the set a if r is a relation on a and there will exist a y belonging to a such that both xy and y z these two belong to r.
00:50
This is because xz belongs to r composition r.
00:53
Now we have already assumed that r is transitive and since r is transitive and x, y, z belong to r, this will imply that xz belongs to r.
01:06
So we can see that for any x z belonging to our composition r, we have that that x z belongs to r.
01:14
So from here we will obtain that our composition r is a subset of r.
01:20
So we have shown one side of the proof.
01:24
Now conversely, conversely, let us show the convert side, conversely led our composition r be a subset of r...