9. [4 Points] In this problem, a bit-string is a sequence of 8 characters selected from 0 and

1. For example, 01011010. Which of the following collections are partitions of the set of bit-strings? Circle your

choices and give an explanation when you think that a set collection isn’t a partition. (a) the set of bit-strings that begin with 1, the set of bit-strings that begin with 00, and

the set of bit-strings that begin with 01. (b) the set of bit-strings that contain the string 00, the set of bit strings that contain the

string 01, the set of bit-strings that contain the string 10, and the set of bit-strings

that contain the string 11. (c) the set of bit-strings that end with 00, the set of bit strings that end with 10, and

the set of bit-strings that end with 11. (d) the set of bit-strings that end with 111, the set of bit-strings that end with 011, and

the set of bit-strings that end with 00. 10. [6 Points] Use the listing method to write the Cartesian products below. Make sure to

use proper notation. (a) {(1, b, c} x 0 (b) {may} X {a,b, 0}

(c) {1,2} x {3,4} x {5,6, 7} 11. [4 Points] Floors and ceilings.

(3) Compute [1.757— [8.051]. (b) Suppose that n is an integer, and m is a real number, and m 7é n. If [m] = n, what

are [—23] and [—m] (express them in terms of n)? 12. [12 Points] In each of the following cases, state whether or not the given function is the

identity ﬁrnct'ion (on its domain). Prove your answers. (a) f1: Z —) Z with rule f1(:1:) = [x+ 1/2]. (b) f2 : Z —) Z with rule f2(a:) = [x— 1/2]. (0) f3 : Z —) Z with rule f3(a:) = Lx/2J + [x/2]. ((1) f4: {0, 1} —) {0, 1} with rule f4($) = [(3: + 1V2].

(e) f5: {0, 1} —) {0, 1} with rule f5(:r) = [2.29/3]. (f) f5: {0,1} —) {0,1} with rule f5(:c) = [m/3l.