CSE 2315 – Discrete Structure HW6 Solution (Ref. Book, 7 th edition) Exercise 5.1 3. 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 x 2 +4y 2 8. a. one-to-one b. many-to-one c. many-to-many d. one-to-many 17. a. symmetric c. reflexive, antisymmetric, transitive 18. a. reflexive, antisymmetric, transitive c. symmetric 19. a. reflexive, transitive b. antisymmetric (because x taller than y and y taller than x is always false, the implication is true), transitive c. reflexive, symmetric, transitive d. antisymmetric (false antecedent) 20. a. antisymmetric, transitive (false antecedent in each case) b. symmetric c. reflexive, symmetric, transitive d. none (not transitive - x ρ y and y ρ x does not imply x ρ x)
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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
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.