3.5
Image of a Set

3.5.1 Introduction

In many areas of mathematics, be it topology, measure theory, real analysis, and others, one seeks to find the image of a set A under the action of a function f : A → B, yielding the image set

equation

In medical imaging, such as CT scans, MRI imaging, X‐rays, one is not interested in images of points, but of images of sets. Although the image viewed by the medical professionals is not the kind of function we have studied thus far, it is a function nevertheless.

Radio graph of axial view (left) and dorsal view (right) of the brain.
Medical imagings are images of sets under some function.

This discussion motivates the image and inverse image of a set.

Diagram of image and inverse image of a set displaying a box labeled A linked by right arrow labeled f to a box labeled f(A) (top) and box labeled f –1(B) linked by left arrow labeled f –1 to box labeled B (bottom).

Figure 3.34 Image and inverse image of a set.

Diagram of image of a set displaying vertical line labeled X with solid circles labeled 1, 2, and 3 indicating A (left) linked with arrows to solid circle markers a and c of vertical line labeled Y (right).

Figure 3.35 Image of a set.

Graph of f-1([2, 5]) = [-2, - 1] ∪ [1, 2] displaying an inverted bell-shaped curve labeled f(x)=x2+1 and broad lines on x-axis points 1 to 2 and -1 to -2 and y-axis points 2 to 5.

Figure 3.36 f−1([2, 5]) = [−2, − 1] ∪ [1, 2].

3.5.2 Images of Intersections and Unions

The following theorem gives an important property for the image of the intersection of two sets.

Diagram displaying an ellipse labeled X enclosing two small overlapping ellipse labeled A and B with three arrows linking to another two overlapping ellipse labeled f(A) and f(B) enclosed by a bigger ellipse labeled Y.

Figure 3.37 Image of an intersection.

Graph displaying an inverted bell-shaped curve labeled y=x2 with arrows marking f(A ∩ B) and f(A) ∩ f(B).

Figure 3.38 Counterexample to show f(A) ∩ f(B) ⊄ f(A ∩ B).

Theorem 1 tells us that set intersections are not preserved under the image of a function. However, if a function is 1–1 then intersections are preserved.

We now see that in contrast to intersections, unions are always preserved under set mappings.

The proof of b) is left to the reader. See Problem 8.

3.5.2.1 Summary

Given a function f : X → Y, where A, B are subsets of X and C, D are subsets of Y, the following properties hold.

  1. f(A ∪ B) = f(A) ∪ f(B)
  2. f(A ∩ B) ⊆ f(A) ∩ f(B) (= if f is 1 ‐ 1)
  3. f−1(C ∪ D) = f−1(C) ∪ f−1(D)
  4. f−1(C ∩ D) = f−1(C) ∩ f−1(D)
  5. f(f−1(C)) ⊆ C (= if f is onto)
  6. A ⊆ f−1(f(A)) (= if f is 1 ‐ 1)
  7. A ⊆ B ⇒ f(A) ⊆ f(B)
  8. C ⊆ D ⇒ f−1(C) ⊆ f−1(D)
  9. images
  10. f−1(C − D) = f−1(C) − f−1(D)
  11. images
  12. images
  13. images
  14. images

Problems

  1. Party Time

    We are having a party with possible desserts

    equation

    where the possible guests are among the group A = {a, b, c, d, e}. Each guest's favorite dessert is indicted by the function f : A → B where

    equation

    What types of desserts will be required if the following groups of guests are invited to the party?

    1. images
    2. images
    3. images
    4. images
    5. f({a, b, c, d, e})
  2. Images of Sets

    Given the sets A = {1, 2, 3, 4},  B = {a, b, c, d} and the function f : A → B defined by

    equation

    find the following:

    1. images
    2. images
    3. images
    4. images
    5. images
    6. images
  3. Interpretation of Images

    Translate the following statements into English. For example y ∈ f(A) means there exists an x ∈ A such that y = f(x).

    1. images
    2. images
    3. images
    4. images
    5. images
    6. images
    7. images
    8. images
  4. Images of Sets

    Given the function f : ℝ → ℝ defined by f(x) = x2 + 2. Find the images of the given sets under the mapping f.

    1. f({−1, 1, 3})
    2. f(Ø)
    3. f([0, 2])
    4. f([−1, 2] ∪ [3, 5])
    5. f−1([−1, 2])
    6. f−1([0, 2])
    7. f−1([6, 11])
    8. f−1([−1, 6])
  5. Continuous Images of Intervals

    Given the continuous function f : ℝ → ℝ defined by f(x) = |x| + 1, find the images.

    1. f([−2, − 1))
    2. f([−2, 3])
    3. f([−2, 2])
    4. f−1([0, 4])
    5. f−1([−2, 0])
    6. f−1({1, 2, 3})
  6. Identity or Falsehood?

    True or false

    equation
  7. Image of a Union

    Show f(A ∪ B) ⊆ f(A) ∪ f(B).

  8. Inverse of Union

    Show f−1(A ∪ B) = f−1(A) ∪ f−1(B).

  9. Complement Identity

    Show images

  10. Composition of a Function with Its Inverse

    Prove the following and give examples to show that equality does not hold.

    1. f[f−1(A)] ⊆ A
    2. A ⊆ f−1[f(A)]
  11. Inverse Images

    Let f : ℕ → ℝ be a function defined by f(n) = 1/n. Find

    1. images
    2. images
    3. images
  12. Inverse Image of an Open Interval

    In topology, a function f is defined as a continuous function if the inverse image of every open set in the range is an open set in the domain. Show that for the function f : ℝ → ℝ defined by f(x) = x2, the inverse image of the following open intervals2 is an open interval or the union of open intervals, and thus f(x) = x2 is a continuous function.

    1. f−1((−1, 1))
    2. f−1((0, 4))
    3. f−1(ℝ)
    4. f−1((4, 16))
  13. Dirichlet's Function

    Given Dirichlet's (shotgun) function3 f : [0, 1] → R is defined by

    equation

    Find

    1. images
    2. images
  14. Image of a Singleton

    Let f : X → Y. Show that for x ∈ X one has

    equation
  15. Connected Sets

    It can be proven that the continuous image of a connected set is connected.4 Find the image of the connected set [−1, 1] under the continuous functions.

    1. f(x) = x3Ans: f([−1, 1]) = [−1, 1]
    2. f(x) = ex
    3. f(x) = 2x + 1
  16. Function of Functions

    Define C[0, 1] to be the set of continuous functions defined on [0, 1].5

    Define

    equation
    1. Find f, g ∈ C [0, 1] so I(f) = 1, I(g) = 0.5, I(h) =  − 4.
    2. Express the following integral property in terms of I
      equation
    3. Is the function I 1–1?
    4. Is the function I onto ℝ?
  17. Internet Research

    There is a wealth of information related to topics introduced in this section just waiting for curious minds. Try aiming your favorite search engine toward properties of images of sets, images, and inverse images of a set.

Notes

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