Page 174 - Discrete Mathematics and Its Applications

2.3 Functions 153

7. Find the domain and range of these functions. a) office.
a) the function that assigns to each pair of positive inte- b) assigned bus to chaperone in a group of buses taking
gers the maximum of these two integers students on a field trip.
b) the function that assigns to each positive integer the c) salary.
number of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 that do d) social security number.
not appear as decimal digits of the integer
18. Specify a codomain for each of the functions in Exercise
c) the function that assigns to a bit string the number of 16. Under what conditions is each of these functions with
times the block 11 appears
the codomain you specified onto?
d) the function that assigns to a bit string the numerical
19. Specify a codomain for each of the functions in Exercise
position of the first 1 in the string and that assigns the
17. Under what conditions is each of the functions with
value 0 to a bit string consisting of all 0s
the codomain you specified onto?
8. Find these values.
20. Give an example of a function from N to N that is
a) 1.1 b) 1.1
a) one-to-one but not onto.
c) −0.1 d) −0.1
e) 2.99 f) −2.99 b) onto but not one-to-one.
1 1 1 1 1
g) + h) + + c) both onto and one-to-one (but different from the iden-
2 2 2 2 2
9. Find these values. tity function).
3 7 d) neither one-to-one nor onto.
a) b)
4 8
3 7 21. Give an explicit formula for a function from the set of
c) − d) −
4 8
e) 3 f) −1 integers to the set of positive integers that is
1 3 1 5
g) + h) ·x a) one-to-one, but not onto.
2 2 2 2
10. Determine whether each of these functions from b) onto, but not one-to-one.
{a, b, c, d} to itself is one-to-one. c) one-to-one and onto.
a) f(a) = b, f(b) = a, f(c) = c, f(d) = d d) neither one-to-one nor onto.
b) f(a) = b, f(b) = b, f(c) = d, f(d) = c 22. Determine whether each of these functions is a bijection
c) f(a) = d, f(b) = b, f(c) = c, f(d) = d from R to R.
11. Which functions in Exercise 10 are onto? a) f(x) =−3x + 4
2
12. Determine whether each of these functions from Z to Z b) f(x) =−3x + 7
is one-to-one. c) f(x) = (x + 1)/(x + 2)
2
a) f (n) = n − 1 b) f (n) = n + 1 d) f(x) = x + 1
5
c) f (n) = n 3 d) f (n) = n/2 23. Determine whether each of these functions is a bijection
13. Which functions in Exercise 12 are onto? from R to R.
14. Determine whether f : Z × Z → Z is onto if a) f(x) = 2x + 1
2
a) f (m, n) = 2m − n. b) f(x) = x + 1
2
2
b) f (m, n) = m − n . c) f(x) = x 3
c) f (m, n) = m + n + 1. 2 2
d) f(x) = (x + 1)/(x + 2)
d) f (m, n) =|m|−|n|.
2
e) f (m, n) = m − 4. 24. Let f : R → R and let f(x) > 0 for all x ∈ R. Show
that f(x) is strictly increasing if and only if the func-
15. Determine whether the function f : Z × Z → Z is onto
tion g(x) = 1/f (x) is strictly decreasing.
if
25. Let f : R → R and let f(x) > 0 for all x ∈ R. Show
a) f (m, n) = m + n. that f(x) is strictly decreasing if and only if the func-
2
2
b) f (m, n) = m + n .
tion g(x) = 1/f (x) is strictly increasing.
c) f (m, n) = m.
d) f (m, n) =|n|. 26. a) Prove that a strictly increasing function from R to it-
e) f (m, n) = m − n. self is one-to-one.
b) Give an example of an increasing function from R to
16. Consider these functions from the set of students in a
discrete mathematics class. Under what conditions is the itself that is not one-to-one.
function one-to-one if it assigns to a student his or her 27. a) Prove that a strictly decreasing function from R to
a) mobile phone number. itself is one-to-one.
b) student identification number. b) Give an example of a decreasing function from R to
c) final grade in the class. itself that is not one-to-one.
d) home town. 28. Show that the function f(x) = e from the set of real
x
17. Consider these functions from the set of teachers in a numbers to the set of real numbers is not invertible, but
school. Under what conditions is the function one-to-one if the codomain is restricted to the set of positive real
if it assigns to a teacher his or her numbers, the resulting function is invertible.