CSE 2315 – Discrete Structure HW6 Solution(Ref. Book, 7th edition)

Exercise 5.13.a. (1, -1), (-3, 3)b. (19,7), (41, 16)c. (-3, -5), (-4, 1/2), (1/2, 1/3)d. ((1, 2), (3, 2))

6.a. x ρ y x > -1

b. x ρ y -2 y 2

c. x ρ y x 2 - y

d. x ρ y x2 +4y2

8.a. one-to-oneb. many-to-onec. many-to-manyd. one-to-many

17.a. symmetricc. reflexive, antisymmetric, transitive

18.a. reflexive, antisymmetric, transitivec. symmetric

19.a. reflexive, transitiveb. antisymmetric (because x taller than y and y taller than x is always false, the implication is true), transitivec. reflexive, symmetric, transitived. antisymmetric (false antecedent)

20.a. antisymmetric, transitive (false antecedent in each case)b. symmetricc. reflexive, symmetric, transitived. none (not transitive - x ρ y and y ρ x does not imply x ρ x)

31.

a. c.

32.a. a is minimal and least c is maximal and greatestb. a and d are minimal b, c, and d are maximalc. is minimal and least {a, c} and {a, b} are maximal

36.a. = {(1,1), (2,2), (3,3), (4,4), (5,5), (1,3), (3,5) (1,5), (2,4), (4,5), (2,5)}

b. ={(a,a), (b,b), (c,c), (d,d), (e,e), (f,f), (a,d), (b,e), (c,f)}

c. ={(1,1), (2,2), (3,3), (4,4), (5,5), (1,2),(2,4), (4,5), (1,4), (1,5), (2,5), (1,3), (3,4), (3,5)}

51.a. {(1,1), (2,2), (3,3), (4,4), (1,2), (2,1), (3,4), (4,3)}b. {(a,a), (b,b), (c,c), (d,d), (e,e), (a,b), (b,a), (a,c), (c,a),(b,c),(c,b), (d,e), (e,d)}

53.reflexive: , x2 - x2 = 0, which is evensymmetric: if x2 - y2=2n then y2 – x2=-2n, which is even.transitive: if x2 - y2=2n and y2 - z2=2m, then x2 - z2=x2 – y2 + y2 - z2=2n + 2m = 2(n+m), which is evenThe equivalence classes are the set of even integers and the set of odd integers.

Exercise 5.23.

6. Minimum time-to-completion is 16 time units. Critical path: 8,3,2,1 or 8,3,7,5,6.

10. For example: 8,3,4,7,5,6,2,1

Exercise 5.49.a. F (the function is not necessarily one-to-one)b. F (every element in the range, not the codomain, must have a unique preimage)c. Td. F

10.a. F (this is part of the definition of any function)b. F (elements in the codomain do not have images)c. T

11.a. not a function from S to T (not a subset of S T)c. function; one-to-one and ontod. not a function from S to T (0 has no associated value)

15.f. bijection; h-1 : ℝ2 ℝ2 where h-1(x, y)=(y-1, x-1)

16.c. not a function (undefined for x=-1, no associated value in R for x<-1)e. not a function (no associated value in ℕ for x=y=0, z=1)

25. For example, f(x)=1/x. For x 1, the value 1/x is greater than 0 but less than or

equal to 1, so f: S T. If f(x1) = f(x2), then 1/x1 = 1/x2 and x1 = x2, so f is one-to-one.Given any value y in T, that is, 0 < y 1, the value 1/y is in S and f(1/y) = 1/(1/y) = y, so f is onto.

33.a. False. Let x = 3.6. Then x = 3 = 3 x.

b. False. Let x = 4.8. Then 2x = 9.6 =9 but 2 x = 2(4) = 8.

48.a. g∘f=12x3 f∘g=48x3

b. g∘f = x2 -2x+1 f∘g=(4x2-1)/2c. g∘f = x f∘g= x

58.a. (1, 6, 4, 8, 3, 5, 2)b. (1,3) ∘ (2,4) ∘ (5,13,6)c. (1, 5, 4, 3, 2)

76.[A] = {A}, [B] = {B, C, F}[D] = {D, E}

Exercise 5.52.For example, n0 = 2, c1 = 1, c2 = 6. For x 2, 1(x3/2) 3x3 – 7x (6)x3/2

Exercise 5.7

2.x = 2, y = 4

6.

Perform the following operations on both A and I :