whenever a E A and Let A and B be non-empty subsets of R with azb bEB. Prove that sup (A) and inf (13) exist and that sup (A) < Could whenever Sup (A) = inf (B) even though a

Let A, B, C be arbitrary finite sets from the same universal set U. - - (a) Is it true that A - B C (A - B) – (B − C)? If "yes", then prove "rigorously"; if "no", then show a concrete counterexample by specifying sets A, B, C where this subset relation does not hold. (To prove an expression of the form MCN "rigorously", you need to consider an arbitrary element x from M and show that x E N.) (b) Is it true that (A - B = A - C) → (B = C)? If "yes", then prove "rigorously"; if "no", then show a concrete counterexample by specifying sets A, B, C where this implication does not hold.

Without proving anything, determine if the following statements are true or false. For any false statements, give an counterexample. (a) A finite, nonempty set of real numbers always contains its supremum. (b) If a < L for every element a in the set A, then sup A < L. (c) If A and B are sets with the property that a < b for every a E A and b e B, then sup A < inf B. (d) If sup A = s and sup B = t, then sup(A+ B) = s+t. Here and elsewhere, the set A+B is defined as A+B = {a+b:a € A, and b e B}. (e) If sup A < sup B then there is an element of B that is an upper bound for A.