MATH 271 Final, Fall 1996

Math. 271, Final Examination, January 29, 1997 P bleml. 18 pts. V ich topics of the course did you like the most ? the least ? Write about those topics a | about your reasons. P tblem 2, 18 pts. Vite what you know about equipotent sets and cardinal numbers. P iblem 3. 12 pts. Vite the definitions of: (« a well-ordered set; (| an ultrafilter; (c a 6-multiplicative family of sets; (c Lim inf An when Apo, Aj, Ae,... are sets. ae Pi »blem 4. 12 pts. / L bye iinet MV ite the statements of: bp tn ® y (« the Cantor diagonal theorem; (1 the Zermelo theorem; (c the Baire theorem; («the Souslin hypothesis. P oblem 5. 10 pts. Li f:NxN—-N_ be given by the formula f(m,n) = m:n. (e Prove that f is a surjection and that f is not an injection. {1 Let A= {0,2,4,...} denote the set of all even natural numbers. Find f7!(A) and f AxA). P sblem 6. 10 pts. Li. X be a set, A be a o-algebra of subsets of X, and jt be a measure on A. (2 Prove that p(BUC)=p2(B)+u(C)—-p(BNC) forevery B,CEA. (tL Let F={BEA:pu(B)>0}. Is F a filter? Why? Pi sblem 7. 10 pts. L. . C denote the set of all sequences (an)S29 such that lim a, =0 and a, €Q for ne 0 noo e ‘hn. What is the cardinality of the set C? Justify your answer! Pi »blem 8, 20 pts. Li, Ao, Ai, Ag,-.. and Bo, Bi, Bo,... be sets. (2 Which inclusion does always hold between the sets (\e29(AnU By) and ( 0 An) U (Neo Bn) ? Prove your claim. (| Give an example when the inclusion in (a) is proper. (« Show that the two sets in (a) are equal when Ap DA, D Ag Dd... and By D By D> Bed... (d) What can be said about the two sets in (a) when Ag C A} C Ay C... and BCB,CBoc...? Good Luck!