CSE 260 Homework 8 Solutions: Integer Division, Modulo Arithmetic, and Base Conversions - , Assignments of Discrete Structures and Graph Theory

Download CSE 260 Homework 8 Solutions: Integer Division, Modulo Arithmetic, and Base Conversions - and more Assignments Discrete Structures and Graph Theory in PDF only on Docsity! CSE 260 Homework 8- Integer Division -Answer 1. Section 2.4: 42 (hint: example 20 on page 159), 46, 47 42. For each pair (i, j)(Z+×Z+) one can order the pairs the same way as in figure 2 of page 159. 46. Part 1: Set of all computer programs in a particular programming lan- guage is countable (problem 45). A programs can be thought of binary strings which is a subset of all possible binary strings.. All binary strings are countable. Therefore, subset is countable. Part 2: Associate the real number 0.d1d2...dn... to the function f with f(n) = dn. For example, if the real is 0.234 . . . then the function will be f(1)= 2 f(2)=3 f(3)=4 and so on. Thus the set of real numbers (between 0 and 1) maps into a subset of the functions. Section 3.4: 27, 28, 31 28: We just calculate using the formula. We are given x0 = 3. Then x1=(4.3+1) mod 7=13 mod 7=6; x2=(4.6+1) mod 7=25 mod 7=4; x3=(4.4+1) mod 7 = 17 mod 7=3. At this point the sequence must continue to repeat 3, 6, 4, 3, 6, 4, . . . forever. 2. Which positive integers less than 30 are relatively prime to 30. {7, 11, 13, 17, 19, 23, 29} 3. The mod function is defined as: mod p : Z → Z(p), mod(x,p)=x mod p, and Z(p) = {0, 1, 2, ...p− 1} We can define add and multiplication in modulo arithmetic as follows: a +p b = (a + b) mod p and a ∗p b = (a.b) mod p, where a, bZ(p) An important modulo arithmetic is in Z(2) as follows: For example, 1+21=(1+1) mod 2=2 mod 2=0 and 1∗21=(1.1) mod 2= 1 mod 2=1