Page 175 - Discrete Mathematics and Its Applications

154 2 / Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

29. Show that the function f(x) =|x| from the set of real inverse of the invertible function f . Notice also that f −1 (S),
numbers to the set of nonnegative real numbers is not the inverse image of the set S, makes sense for all functions f ,
invertible, but if the domain is restricted to the set of non- not just invertible functions.)
negative real numbers, the resulting function is invertible. 42. Let f be the function from R to R defined by
2
30. Let S ={−1, 0, 2, 4, 7}. Find f(S) if f(x) = x . Find
a) f(x) = 1. b) f(x) = 2x + 1. a) f −1 ({1}). b) f −1 ({x | 0 <x < 1}).
2
c) f(x) = x/5 . d) f(x) = (x + 1)/3 . c) f −1 ({x | x> 4}).
2
31. Let f(x) = x /3 . Find f(S) if 43. Let g(x) = x . Find −1
−1
a) S ={−2, −1, 0, 1, 2, 3}. a) g ({0}). b) g ({−1, 0, 1}).
c) g −1 ({x | 0 <x < 1}).
b) S ={0, 1, 2, 3, 4, 5}.
c) S ={1, 5, 7, 11}. 44. Let f be a function from A to B. Let S and T be subsets
of B. Show that
d) S ={2, 6, 10, 14}.
a) f −1 (S ∪ T) = f −1 (S) ∪ f −1 (T ).
32. Let f(x) = 2x where the domain is the set of real num- b) f −1 (S ∩ T) = f −1 (S) ∩ f −1 (T ).
bers. What is
45. Let f be a function from A to B. Let S be a subset of B.
a) f(Z)? b) f(N)? c) f(R)? −1
Show that f (S) = f −1 (S).
33. Suppose that g is a function from A to B and f is a 46. Show that x + is the closest integer to the number x,
1
function from B to C. 2
except when x is midway between two integers, when it
a) Show that if both f and g are one-to-one functions, is the larger of these two integers.
then f ◦ g is also one-to-one. 1
47. Show that x − is the closest integer to the number x,
2
b) Show that if both f and g are onto functions, then except when x is midway between two integers, when it
f ◦ g is also onto. is the smaller of these two integers.
∗ 34. If f and f ◦ g are one-to-one, does it follow that g is 48. Show that if x is a real number, then x −xx = 1if x
one-to-one? Justify your answer. is not an integer and x −xx = 0if x is an integer.
∗ 35. If f and f ◦ g are onto, does it follow that g is onto? 49. Show that if x is a real number, then x − 1 < x ð x ≤
Justify your answer. x <x + 1.
2 50. Show that if x is a real number and m is an integer, then
36. Find f ◦ g and g ◦ f , where f(x) = x + 1 and g(x) =
x + 2, are functions from R to R. x + m = x + m.
51. Show that if x is a real number and n is an integer, then
37. Find f + g and fg for the functions f and g given in
Exercise 36. a) x< n if and only if x <n.
b) n<x if and only if n< x .
38. Let f(x) = ax + b and g(x) = cx + d, where a, b, c,
and d are constants. Determine necessary and suffi- 52. Show that if x is a real number and n is an integer, then
cient conditions on the constants a, b, c, and d so that a) x ≤ n if and only if x ð n.
f ◦ g = g ◦ f . b) n ≤ x if and only if n ≤ x .
39. Show that the function f(x) = ax + b from R to R is
53. Prove that if n is an integer, then n/2 = n/2if n is even
invertible, where a and b are constants, with a = 0, and
and (n − 1)/2if n is odd.
find the inverse of f .
54. Prove that if x is a real number, then −x =− x and
40. Let f be a function from the set A to the set B. Let S and −x =− x .
T be subsets of A. Show that
55. The function INT is found on some calculators, where
a) f(S ∪ T) = f(S) ∪ f(T ). INT(x) = x when x is a nonnegative real number and
b) f(S ∩ T) ⊆ f(S) ∩ f(T ). INT(x) = x when x is a negative real number. Show
that this INT function satisfies the identity INT(−x) =
41. a) Give an example to show that the inclusion in part (b)
in Exercise 40 may be proper. −INT(x).
56. Let a and b be real numbers with a< b. Use the floor
b) Show that if f is one-to-one, the inclusion in part (b)
in Exercise 40 is an equality. and/or ceiling functions to express the number of inte-
gers n that satisfy the inequality a ≤ n ≤ b.
Let f be a function from the set A to the set B. Let S be a
subset of B. We define the inverse image of S to be the subset 57. Let a and b be real numbers with a< b. Use the floor
of A whose elements are precisely all pre-images of all ele- and/or ceiling functions to express the number of inte-
ments of S. We denote the inverse image of S by f −1 (S),so gers n that satisfy the inequality a <n<b.
f −1 (S) ={a ∈ A | f(a) ∈ S}.(Beware: The notation f −1 is 58. How many bytes are required to encode n bits of data
used in two different ways. Do not confuse the notation intro- where n equals
duced here with the notation f −1 (y) for the value at y of the a) 4? b) 10? c) 500? d) 3000?