Prove that is a transitive set if and only if is a transitive set.
I think I was able to prove one direction.
Suppose is transitive. We need to show that . Let . Then . Since is transitive, we know . Hence, .
For the other direction. I need to show that . I have a hard time of effectively using the hypothesis that is a transitive set. Can someone give me a hand here? Thanks.
Thank you very much for your help Alice. We did prove the lemmas in class, one more lemma we proved is that if is transitive then . I didn't approach this problem as you did, which is why I was stuck I think. I was trying to show that any element in is also an element in by the usual way, but I couldn't get anywhere.