Discrete Math 9 questions, please do them all with steps! Thank you and good luck!

4. (a) Prove that [55] = [a:] and [113] = {55] . (b) Give a proof by cases that [433] = [as] + [so + i] + [w + a + [as + i] . 5. (a) Use the Euclidean algorithm to determine whether or not the years 1812 and 2013 are relatively prime. (b) Let k,m,n E I\7. (a) Without computing the value of 100! , determine how many zeros are at the end of this number when it is written in decimal form. Justify your answer. (b) Find all solutions to m2 n2 = 105, for which both m and n are integers. Hint: Both proofs rely on the Factorization Theorem. 8. (a) Suppose that Hilbert's Grand Hotel is fully occupied, but the hotel closes all the even numbered rooms for maintenance. Show that all guests can remain in the hotel. (b) Show that a countably innite number of guests arriving at Hilbert's fully occupied Grand Hotel can be given rooms without evicting any current guest. 9. Determine whether each of these sets is countable or uncountable. For those that are countably innite, exhibit a oneto-one corre5pondence between the set of positive integers and that set. (a) integers not divisible by 3 (b) integers divisible by 5 but not by 7 (c) the real numbers with decimal representations consisting of all Is (d) the real numbers with decimal representations of all Is or 9s 1 . Let the function f : R _ I be given by f ( ` ) = = + 1 2 - 1 if * * 1 , f ( * ) = 1 if 2 = 1 . Draw the graph of I versus the values of I . Is fa bijection ( i.e., one - to- one and onto ) ? If yes then* give a proof and derive a formula for F - 1 . If no then explain why not . 2. Let f : ZZ - 2 2 be defined as f ( m, n ) = ( m - 17 , 12 ) . Is f indeed a properly defined function from* 2 2 to 2 2 ? Is fa bijection , i.e., one - to- one and onto ? If yes then give a proof and derive a formula for f - 1 . If no then explain why not . Also derive a formula for the composite function f., for KEZ * . Here Iz denotes the composite function fof , Is denotes the composite function fo fo f , etc . ( You are asked to derive the formula for ${ for general KE Z * . ) Is It a bijection ? If yes then give a proof and derive a formula for its inverse f. . . If no then explain why not . 3 . If A and Bare sets and f : A - B, then for any subset S of A we define f ( S ) = ( b E B : 6 = f ( a ) for some at .S ] Similarly , for any subset I of B we define the pre-image of I as f - ! ( I ) = [a EA : f ( a ) EI ) . Note that f - ! ( I ) is well defined even if f does not have an inverse ! Now let f : R _ K be defined as f ( [ ) = 12. Let SI denote the closed interval [ - 2 , 1] , that is all I ER that satisfy - 2 { I S 1 , and let S 2 be the open interval ( - 1 , 2 ) , that is all I E R that satisfy* - 1