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
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.
This discussion motivates the image and inverse image of a set.
The following theorem gives an important property for the image of the intersection of two sets.
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.
Given a function f : X → Y, where A, B are subsets of X and C, D are subsets of Y, the following properties hold.
We are having a party with possible desserts
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
What types of desserts will be required if the following groups of guests are invited to the party?
Given the sets A = {1, 2, 3, 4}, B = {a, b, c, d} and the function f : A → B defined by
find the following:
Translate the following statements into English. For example y ∈ f(A) means there exists an x ∈ A such that y = f(x).
Given the function f : ℝ → ℝ defined by f(x) = x2 + 2. Find the images of the given sets under the mapping f.
Given the continuous function f : ℝ → ℝ defined by f(x) = |x| + 1, find the images.
True or false
Show f(A ∪ B) ⊆ f(A) ∪ f(B).
Show f−1(A ∪ B) = f−1(A) ∪ f−1(B).
Show
Prove the following and give examples to show that equality does not hold.
Let f : ℕ → ℝ be a function defined by f(n) = 1/n. Find
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.
Given Dirichlet's (shotgun) function3 f : [0, 1] → R is defined by
Find
Let f : X → Y. Show that for x ∈ X one has
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.
Define C[0, 1] to be the set of continuous functions defined on [0, 1].5
Define
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.
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