3.3: Reverse Relations and Images

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We begin this section with a review of the term "Image". The first appearance of the term "Image" was in Section 3.1 when we discussed the topic: Functions. Now, we are going to elaborate on this term a little bit more. We begin with following example.

Example

Let A be the set of all men and B be the set of all women. A relation $LaTeX: f:A\longrightarrow B$ is defined as "For each element x in A, assign an element y in B to x if y is the biological mother of x".

The set A is called the domain of this relation and the set B is called the codomain of the relation. From the previous section, Section 3.2, we know this relation

is a function because for each x in A, there is at most one y in B that can be assigned to x (For each man, there is only one biological mother)
is total because for each x in A, there will be one y in B that is assigned to x (Each man will have a biological mother)

But this relation is not surjective. To be surjective, for each element y in B, there must be at lease one element x in A such that y is assigned to x. That means for each woman in B, she has to be a biological mother of someone in A. Obviously, this is not true because some women may not have a son.

Now, the set B (the set of all women) can be divided into two subsets: one that contains all the women who have a son and the other one contains all the women who don't have a son. We use S to represent the set that contains all the women who have a son, then S is called the image of this relation, and we write it as R(A) = S or S = R(A).

An image of a relation is a subset of the codomain. It can be the same as the codomain if all the elements in the codomain can be assigned to an element in the domain. If this is the case, then the relation is surjective.

Example

Let A be the set of all real numbers and B also be the set of all real numbers. A relation $LaTeX: f:A\longrightarrow B$ is defined as "For each element x in A, assign an element y in B to x if y = x²"

Both domain and codomain of this relation is the set of all real numbers. This relation

is a function because, for each element x in A, there is at most one element in B that can be assigned to x
is total because for each element x in A, there will be an element y in B such that y = x²

But, it is not surjective because not every y in B can be assigned to an element in A. In other words, for some element y, we can't find an x in A such that y = x². For example, for y = -1, there isn't anything in A that will make -1 = x² be a true statement. Again, the codomain is divided into to sets: one that contains all the positive numbers y in B for which we can find an x in A such that y=x² . We use S to denote this set. The other set contains all the negative numbers y in B for which, we can't find a number x in A such that y = x².

In this example, the subset S is the image of the relation. From algebra, we know the set S consists of all the numbers that are greater than or equal to 0. That is S = {x $LaTeX: \in$ B : x $LaTeX: \ge$ 0}

With the help of those two examples, hope you have grasped the concept of "image" for a given relation.

Now, let's tackle the concept of "Inverse Relation"

Definition

Let A and B be sets. A relation $LaTeX: f:A\longrightarrow B$ assigns an element y in B to an element x in A. The inverse of this relation is also a relation which is denoted by ^-1: $LaTeX: B\longrightarrow A$ and is defined as "Assign an element y in A to an element x in B if x was assigned to y in the relation $LaTeX: f:A\longrightarrow B$

Such precise mathematical/logical definition is always very difficult to understand. It is like those laws written in a law book. It is not really for the normal people like you and me to read. Therefore, we need interpretations and examples to help us to understand the concepts. So are those mathematical/logical definitions.

Example

Let A be a set {a, b, c, d, e} and B be a set {3, -1, 2, 4}. A relation $LaTeX: f:A\longrightarrow B$ is defined by following table:

preview

Then the inverse of this relation $LaTeX: f^{-1}:B\longrightarrow A$ will be defined as

preview_ozeb

From this example, we can see to get an inverse of a relation, all we need to do is switch the domain to codomain and switch the codomain to domain and at the meantime, reverse the directions of those arrows.

It is going be even easier to get an inverse relation if we use a set of ordered pairs to represent a relation.

In this example, we can use following set of the ordered pairs to represent the original relation $LaTeX: f:A\longrightarrow B$

LaTeX: f : {(a, -1), (b, 3), (b, 2), (c, 2), (d, 4), (e, 4)}

To get the inverse of this relation, all we need to do now is to switch the order of two elements in each ordered pair:

$LaTeX: f^{-1}$ : {(-1, a), (3, b), (2, b), (2, c), (4, d), (4, e)}

From this example we can also see the original relation $LaTeX: f:A\longrightarrow B$ is a function, but its inverse is not.

Example

Now, the inverse of this relation $LaTeX: f^{-1}:B\longrightarrow A$ is defined as "For each element x in B (each woman in B), assign an element y in A (a man in A) to x if y is a son of x". We can see the original relation is a function, but its inverse is not. The original relation is not surjective, but its inverse is. The original relation is total, but its inverse is not.

Example

The inverse of this relation $LaTeX: f^{-1}:B\longrightarrow A$ will be defined as "For each element x in B, assign an element y in A to x if y² = x" For example, for the element x=16 in B, we will assign the number y=4 in A to it because 4² = 16.

This last example may be a challenge. Actually, it is not too bad. The original relation maps each number in the domain to its square and the inverse relation maps each number in the domain to it square root. The original relation is function, but its inverse is not because each number in the domain have two different square roots: one is positive and the other is negative.

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