1. Fix some set X. In this question we we will show how C is 'kind of like\" g by talking about \"upper bounds\" and \"lower bounds" of collections of subsets of X. Recall that the powerset WHO is the set of all subsets of X. C E 35(X) means that C C X. Furthermore, a subset A C c95(X) has elements of the form B E A= where B C X. In other words, A is a collection of sets. (a) Let C E ?(X). and let A C 33(X). \"Te say C is an upper bound ofA if B C C for all 3 E A. Given Bth E 33[X), show that Bl U 5'2 E 555(X) is an upper bound of {31.32} C .9300. we say an upper bound 5' of A C 95(X) is a least upper bound if for all upper bounds C of A: S C C. Given 3th E .S'NX)= show that if C is an upper bound of {31,32}. then Bl U Bg C C. Use this with your result in (a) to show that Bl U 82 is the unique least upper bound of {81.32}. (Him: For the second part, rst show that Bl U B2 is a least upper bound. Then= show that if S is another least upper bound, 5' = Bl U 32) 'Write down a corresponding denition for some C E 395(X) to be a lower bound of A C 93(X). Use it to prove that Bl n 32 is a lower bound of {81, BE}. Show that @(X) an analogue to the least upper bound property: evervv non empty collection A C 355(X) has a least upper bound S. You may nd the following to be helpful: given a collection of sets A C ?(X), you can dene a set which is the union over this collection= UB:={wEX::rEBforsomeBEA}E33(X) BEA