and 
} 
}.
1.1(f) {{ and } }.
1.4(d) 
.
1.8(b) 
, since 
for all n.
1.8(h) 
and 
.
1.22 (e) It is one-to-one. For if if p(x) = p(y), then the fraction part of x must equal that of y.
Hence their integer parts must also be equal.
The range of p is the set of nonnegative real numbers whose integer part are even.
For let x = i + f, where i is the integer part of x and f is the fraction part of x.
Then p(x) = 2i + f because the floor of x is equal to i. So the integer part of p(x) is 2i
and the fraction part is f.
1.57(b) 
?
Let 
. Then 
or 
.
If , then 
and y = f(x).
Hence 
and y = f(x).
Hence 
.
Similarly for the case .
Hence .
1.23 f is onto but g is not necessarily onto.
g is one-to-one but f is not necessarily one-to-one.
1.24(a) There are altogether eight of them, for example
g(7) = 2, g(8) = 4, g(9) = 3, g(10) = 6.