(i) Prove that .
That is not true!
This is true.
(i) Prove that .
(1) Let be a function and . (i) Prove that . Prove that the converse is not universally true. Give a simple condition on which is equivalent to the converse. (ii) Prove that . Prove that equality is not universally true. (iii) Prove that .
(i) So . Then . Thus . The converse is . Suppose that and . Let . Then but . Is this correct? What would be the condition on which is equivalent to its converse?
(ii) which is a subset of . What would be a counterexample to show inequality?
(iii) This is basically the same as part (ii) except with an ?
Thanks
Do you see how different in detail my example is from the one you tried to give? To be honest, I would not accept it. It lacks detail that shows that you really understand how functions operate. But I am not grading you. So I cannot say if it correct or not; just I do not find it so.
I have always advised students to simply copy out proofs from well-written textbooks, a book at the correct level. Set theory & logic is the basis of all proofs.
Well it certainly is not meant to be approached in a shotgun way that your assortment of problems indicates you are trying.
My own take on this is that Math is learned in basically the same manner as practically any other field. You need to work problems, ask questions, and work more problems. Some time for the information to "gel" helps, as well as seeing how the material applies to either real-life problems or how it is applied to other fields.
Just keep at it and be patient. You'll get it.
-Dan
Maybe the problem is that this is very abstract to thee. The first book on math having serious proofs I ever read was on number theory which is much less abstract. And perhaps this will make you understand what proofs are about. All set theory is is doing it much much more formally.