Exercises
241
20. Prove that the function F : Z -* Z defined as F(n) = n + 6 is a bijection.
21. For each of the following functions, prove that the function is 1-1 or find an appropriate pair of points to show that the function is not 1-1:
(a) F 2 Z- Z
F) n 2
for n > 0
F I)=-n2 for n < 0
(b) F :R
RJR
F(x)= x +l
forxEQ
12x
for xQ
(c) F :R -R J
\
+3x+2 forxEQ
F(x)= x3
forxgQ
(d) F Z -Z 2
n +1 fornodd
Fln
I n3
for n even
22. (a) Find functions from R to R that are:
i. strictly decreasing
ii. decreasing but not strictly decreasing
iii. neither increasing nor decreasing
iv. both increasing and decreasing
(b) Show that no F : -+ R is both increasing and strictly decreasing.
(c) Find a subset X C JR and a function F : X -+ X where F is both strictly increasing and strictly decreasing.
23. Construct functions with the following properties:
(a) F : N -+ N such that range(F) = N and, for each n E N, there exist exactly two
solutions for the equation F(x) = n.
(b) F : N -> N such that, for each n E N, there are exactly n solutions for the equation
F(x) = n.