Example 3 For the function f : X →Y defined by f : x→sin x , identify the
domain, range, image of f , and the preimage of [0,1] . See Figure 4
Solution Both the domain and codomain of f are the real numbers , the range
of f is [−1,1] , and the preimage of [0,1] is
Note that the image of f is only a subset of the codomain.
Sine graph
Figure 4
Although we have seen that intersections are not always
preserved
under the action of a function unless the function is one-to-one, unions of
sets are always preserved .
Theorem 3 Let f : X →Y . If
 X , then
unions are preserved.
That is
|
Proof: We begin by showing
If 
this means there
exists an 
such
that 
.
Hence, 
or 
and so 
or 
, and
so we
have 
which verifies
. We
leave
the verification that 
to the reader. See Problem 4.
Although the intersection of sets is not preserved for functions, it is
preserved for the inverse of a function.
Theorem 4 Let f : X →Y and
 , then
the inverse image of
intersections and unions satisfies
 |
Proof: We prove a) by the series of equivalent statements :
The proof of b) is left to the reader. See Problem 5.
Summary :
Given a function f : X →Y where A,B are subsets of X , and C,D are subsets
of Y , we have the fol lowing properties . Notice how the inverse image always
preserves
unions and intersections, although not always true for the image of a function.
Problems
5. Translate the following statements. For example y∈ f ( A) means there
exists an x∈ Asuch that y = f (x) .
6. Let 
. Find the following.
7. Let f : A−{ 0 }→B be defined by
. Find
8. Let f : A→B where A = {1,2,3, 4},B = {a,b,c, d}. If
f = {(1, a), (2,a) , (3,c), (4,b)}
find the following.
9. (Identity or Falsehood?) Let f : A→B and let X ,Y be
subsets of A .
Prove or disprove
10. (Families of Sets) Let f : X →Y and
, i∈I is a
family of subsets
of X . Prove the following.
11. (Image of a Union) Show
.
12. (Inverse of Union) Show 
.
13. (Compliment Identity) Show 
14. (Composition of a Function with Its Inverse) Show the following and give
examples to show we do not have to prove equality.
15. Let f : N→ R be a function defined by f (n) =1/ n . Find
16. (Classroom Puzzle) Let A be the set of students in
your Intro to
Abstract Math Class and B be the natural numbers from 1 to 100. Suppose
now we as sign to each person in the class the age of that person. That is, if
x is a student, then , where n is the age of x . If we now assign to each
natural number n in f ( A) those students whose age is n . Under what
conditions will this be a function from f ( A) to A .
17. (Inverse Image of an Open Interval) In topo logy , a continuous function f
is defined to be a function such that the inverse image of every open set is an
open set. Show that for the continuous function f : R→R defined by
, the inverse image of the following open intervals2 is an open
interval or the union of open intervals.
18. (Dirichlet’s Function) Dirichlet’s function f :
[0,1]→R is defined by by
Find
19. Properties of Images One can prove for a continuous
function f : R→R ,
the following properties hold:
a) the image f ([a,b]) of a closed interval [a,b] is not necessarily a
closed interval.
b) The image f ( A) of a bounded set A is not necessarily bounded.
c) The image of a closed and bounded interval is closed and bounded.
d) The inverse image 
of an open interval is an open set; either
an open interval or the union of open intervals.
Give examples of a function f and domains where these properties hold.
20. Let f : X →Y . Show that for x∈ X one has f ({x}) = { f (x)}.
21. (Connected Sets) It can be proven that the continuous image of a
connected set is connected. Show that the image of the connected set (−1,1)
under the functions.
