5. Let A and B be compact subsets of R. (a) Prove that AnB is compact (b) Prove that AUB is compact. (c) Find an infinite family An of compact sets for which UAn is not compact. o-f (d) Suppose that An is a compact set for n 21. Prove that An is compact.
16 pts) PROBLEM 21. Let f:X →Y be a function, let Xi, X2SX andlet Yi, ½SY. ) Write down the definitions of f(Xi) and f (Y 。ín½) = f-'(%) nf-106). (ii) Prove that (ii) Prove that f(XinX)(xnf(xa) (i) Find a counterexample to the statement (xinx) J(x)n(X) Do not show how you found the right ideas. Present detailed and carefully set out definitions and proofs only END of PROBLEM 21 16 pts) PROBLEM 21. Let f:X →Y be a function, let...
2.4. Prove AN(BUC) -An B)U(ANC) P(AUB) = PCA)+PCB)- P(ANB)
Prove equalities involving sets A, B, C and D a) (AIB)U(C1B) = (AUC) IB b) (AUB)-(ANB) = (A-8)U(-A) c) (AxB) OLC xD) - (ANC) x (BND) d) (AXB) (BAA) = (ANB)X(AMB)
Prove or disprove the following: If f(n) =O(g(n)) then nf(n) = O(ng(n))
QUESTION 1 (3 pts). Prove Theorem 2.3.6(f). That is, fix a family of norms on F"n, n > 1, and prove that for all A E Mm,n(F) and Be Mn,k(F), we have ||AB|| < ||A|| || B||. QUESTION 1 (3 pts). Prove Theorem 2.3.6(f). That is, fix a family of norms on F"n, n > 1, and prove that for all A E Mm,n(F) and Be Mn,k(F), we have ||AB||
4 [8 pts. Consider sets A, B, and X. Recall 2 is the powerset of set Y. Prove the following (XC A)Л (х Св) (а) XСАП В [Recall to prove a biconditional statement like S T S T, you have to prove both S T, and (b) 2(AnB) 24n 2B equal could prove t E X = teY] Hint: Use part (a). Also recall that to prove sets X and Y are we 4 [8 pts. Consider sets A, B,...
4. Consider a function f : X → Y. 4a) (5 pts) Let C, D be subsets of Y. Prove that f (CND)sf1(C)nf-1(D). 4b) (10 pts) Let A, B be subsets of X and assume the function f be one-to- one. Prove that f(A) n f(B)Cf(An B) (Justify each of your steps.) 4c) (4pts) Find an example showing that if the function f is not one-to-on the inequality (1) is violated.
2. Prove the following Theorems: (a). Prove that the real line with the standard topology is Hausdorff. (b). Prove that int(ANB) = int(A) n int(B) Y is a homeomorphism. Then if X is a (c). If X and Y are topological spaces and f: X Hausdorff space then Y is Hausdorff. (d). Theorem 4.2
prove that A is non singular 5.(25 pts) For each positive integer n, let f()(+2)(1)(0,1. Let f()-0, (1) Prove that (fn) converges to fpointwisely on (0, 1) (2) Does (n) converges to f uniformly on (0, 1]?