Discrete Math Assignment 1.docx

Assignment 1 1) Exercise Set 1.1 (p. 5+ of Epp, fifth edition): Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when it is divided by 6? a. Is there an integer n such that n has___? So, the statement becomes “Is there an integer n such that n has a remainder of 2 when it is divided by 5 and a remainder of 3 when it is divided by 6 ?” b. Does there exist ___such that if n is divided by 5 the remainder is 2 and if___? So, the statement becomes “Does there exist an integer n such that if n is divided by 5 the remainder is 2 and if n is divided by 6 the remainder is 3? 2) Exercise Set 1.2 (p. 14+): a. Is 2 [ {2}? yes b. How many elements are in the set {2, 2, 2, 2}? one c. How many elements are in the set {0, {0}}? two d. Is {0} [ {{0}, {1}}? yes e. Is 0 [ {{0}, {1}}? no A. Is 3 E {1, 2, 3}? Yes b. Is 1 C {1}? no c. Is {2} E {1, 2}? no d. Is {3} E {1, {2}, {3}}? yes e. Is 1 E {1}? yes f. Is {2} C {1, {2}, {3}}? no g. Is {1} C {1, 2}? yes h. Is 1 E {{1}, 2}? no i. Is {1} C {1, {2}}? yes j. Is {1} C {1}? yes

Let S = {2, 4, 6} and T = {1, 3, 5}. Use the setroster notation to write each of the following sets, and indicate the number of elements that are in each set. a. S X T {(2*1), (2*3), (2*5), (4*1), (4*3), (4*5), (6*1), (6*3), (6*5)} b. T X S {(1*2), (1*4), (1*6), (3*2), (3*4), (3*6), (5*2), (5*4), (5*6)} c. S X S {(2*2), (2*4), (2*6), (4*2), (4*4), (4*6), (6*2), (6*4), (6*6)} d. T X T {(1*1), (1*3), (1*5), (3*1), (3*3), (3*5), (5*1), (5*3), (5*5)} 3) Exercise Set 1.3 (p. 22+): 2) Let C = D = {-3, -2, -1, 1, 2, 3} and define a relation S from C to D as follows: For every (x, y) E C X D, (x, y) E S means that 1/X- 1/Y y is an integer. How do we define relationship S from C to D? For all (X, Y) E C x D X S Y if 1/X- 1/Y Is an Integer. Relation written symbolically as follows: X S y Means that (x, y) E S a. Is 2 S 2? x=2, y=2 2 S 2 since (½) – (½)= 0 is an integer. Is -1 S -1? X=-1, Y=-1 -1S-1 Since (-1/2) -(1/2) = 0 is an integer. Is (3, 3) E S? X=3 y=3 1/3 – 1/3= 0 is an integer. Is (3, -3) E S? X=3 y=-3 1/3 – (1/-3) = 2/3 is not an integer b. Write S as a set of ordered pairs. S= {(x, y) E S|1/Y 1/Y Is an integer

Sets: C = D = {-3, -2, -1, 1, 2, 3} Substitute x =1 and y=1 in 1/x= 1/y is an integer. 1/1-1/1=0 is an integer. Substitute x=1 and Y=-1 1/1+1/1=2 is an integer Substitute x=-1 and y=1 -1/1-1/1=-2 is an integer Substitute x=-1 and y=-1 -1/1+1/1=0 is an integer Substitute x=2 and y=2 ½- ½ =0 is an integer Substitute x=2 and y =-2 ½ + 1/2 = 1 is an integer Substitute x= -2 and y=2 1/-2 – ½ =-1 is an integer Substitute x=-2 and y =-2 1/-2 +1/2=0 is an integer Substitute x= 3 and y =3 1/3-1/3= 0 is an integer Substitute x=-3 and y=-3 1/-3+1/3=0 is an integer The set of ordered pairs: S= {(1,1), (1, -1), (2,2), (3,3, (-1, -1), (-1,1), (2, -2), (-2,2), (-2, -2), (-3, -3)} c. Write the domain and co-domain of S. Domain C={-3,-2,-1,1,2,3} and co-domain = D= {-3,-2,-1,1,2,3} d. Draw an arrow diagram for S.

C S D 6) Define a relation R from R to R as follows: For every (x, y) [ R E R, (x, y) [ R means that y 5 x 2 . X R Y if y=x^2 A. Is (2, 4) E R? Yes, (2,4) E R as 4=2^2 Is (4, 2) E R? No, (4,2) E R as 2 not 4^2. Is (-3) R 9? Yes, -3 R 9 as 9= (-3) ^2 Is 9 R (23)? No, because -3 not 9^2 b. Draw the graph of R in the Cartesian plane. y 3 2 1 1 -1 -2 -3 3 2 -1 -2 -3 1 2 -1 -2  Y=x^2 1 2

10) Find four relations from {a, b} to {x, y} that are not functions from {a, b} to {x, y}. To not be a function, it must either have no elements or multiple elements with some given first component (a or b). 1) For R = {(a, x), (a, y), (b, x)} is not a function from {a, b} to {x, y} because there are two values related to a 2)R = {(a, x)} is not a function from {a, b} to {x, y} because there are no values related to b 3)R=(a, y) 4)R=(b,x) 5)R=(b,y) 12) Let A = {x, y} and let S be the set of all strings over A. Define a relation C from S to S as follows: For all strings s and t in S, (s, t) E C means that t = ys. Then C is a function because every string in S consists entirely of x’s and y’s and adding an additional y on the left creates a single new string that consists of x’s and y’s and is, therefore, also in S. Find C(x) and C(yyxyx). 1. C(X): We input one string ‘x’ A ‘y’ must be added at the start pf the string in order to have a relation to C. By combining ‘y’ and ‘x’ we create a string yx. 2. C(yyxyx): Input ‘’yyxyx’ We apply C: “yyyxyx” We should start by reapplying the relation to C like we did in prior steps, and this tells us to add another “y” Input: " yyyxyx " Apply C: " yyyyxyx " After the second application of C, we get " yyyyxyx ."

As a result, C(yyxyx) = yyyyxyx 15) Let X 5 {2, 4, 5} and Y 5 {1, 2, 4, 6}. Which of the following arrow diagrams determine functions from X to Y? From the arrow diagram option “D” is the answer. This is a function with Domain = {2,4,5} Co-domain ={2,6} 4) Exercise Set 1.4 (p. 35+) 4) Graph H has vertex set {v1, v2, v3, v4, v5} and edge set {e1, e2, e3, e4} with edge-endpoint function as follows :

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