Inverse Relation

An inverse relation, as its name suggests, is the inverse of a relation. Let us recall what is a relation. A relation is the collection of ordered pairs. Let us consider two sets A and B. Then the set of all ordered pairs of the form (x, y) where x ∈ A and y ∈ B is called the cartesian product of A and B, which is denoted by A x B. Any subset of this cartesian product A x B is a relation. Then what is an inverse relation of a relation? Do you think that it is set of all ordered pairs that are obtained by interchanging the elements of the ordered pairs of the original relation? Then yes, you are right!

Let us explore more about the inverse relation in different cases, its domain, and range along with a few solved examples. Also, we will see what is inverse relation theorem along with its proof.

An inverse relation is the inverse of a relation and is obtained by interchanging the elements of each ordered pair of the given relation. Let R be a relation from a set A to another set B. Then R is of the form {(x, y): x ∈ A and y ∈ B}. The inverse relationship of R is denoted by R^-1 and its formula is R^-1 = {(y, x): y ∈ B and x ∈ A}. i.e.,

• The first element of each ordered pair of R = the second element of the corresponding ordered pair of R^-1 and
• The second element of each ordered pair of R = the first element of the corresponding ordered pair of R^-1.

Inverse Relation Definition

In simple words, if (x, y) ∈ R, then (y, x) ∈ R^-1 and vice versa. i.e., If R is from A to B, then R^-1 is from B to A. Thus, if R is a subset of A x B, then R^-1 is a subset of B x A.

Inverse Relation Examples

Have a look at the following relations and their inverse relations on two sets A = {a, b, c, d, e} and B = {1, 2, 3, 4, 5}.

• If R = {(a, 2), (b, 4), (c, 1)} ⇔ R^-1 = {(2, a), (4, b), (1, c)}
• If R = {(c, 1), (b, 2), (a, 3)} ⇔ R^-1 = {(1, c), (2, b), (3, a)}
• If R = {(b, 3), (c, 2), (e, 1)} ⇔ R^-1 = {(3, b), (2, c), (1, e)}

The domain of a relation is the set of all first elements of its ordered pairs whereas the range is the set of all second elements of its ordered pairs. Let us consider the first example from the list of above examples and find the domain and range of each of the relation and its inverse relation.

• For R = {(a, 2), (b, 4), (c, 1)}, domain = {a, b, c} and range = {2, 4, 1}
• For R^-1 = {(2, a), (4, b), (1, c)}, domain = {2, 4, 1} and range = {a, b, c}

What did you observe here? Aren't the domain and range interchanged for R and R^-1? i.e.,

• the domain of R^-1 = the range of R and
• the range of R^-1 = the domain of R.

Note: If R is a symmetric relation (i.e., if (b, a) ∈ R, for every (a, b) ∈ R), then R = R^-1. For example, consider a symmetric relation R = {(1, a) (a, 1), (2, b), (b, 2)}. The inverse of this relation is, R^-1 = {(a, 1), (1, a), (b, 2), (2, b)}. Technically, R = R^-1, because the order of elements is NOT important while writing a set. In this case, the domain of R = range of R = domain of R^-1 = range of R^-1 = {1, 2, a, b}.

If a relation is given as a graph, then its inverse is obtained by reflecting it along the line y = x. This is because the inverse of a relation is nothing but the interchanged ordered pairs of the given relation. To graph the inverse of a relation that is given by a graph,

• Choose some points on the given relation (graph).
• Interchange the x and y coordinates of each point to get new points.
• Plot all these new points and join them by a curve which gives the graph of the inverse relationship.

You can see a relation R that is represented by a circle in the second quadrant, some points on it which are transformed into new points (the transformations are showed by dotted lines) by interchanging the x and y coordinates, and the inverse relation R^-1 that is represented by a circle in the fourth quadrant in the figure below.

If a relation is given in algebraic form, like R = {(x, y): y = 3x + 2}, then its inverse is found using the following steps.

• Interchange the variables x and y.
  In the above example, if we interchange x and y, we get x = 3y + 2
• Solve the above equation for y.
  In the above example, x - 2 = 3y ⇒ y = (x - 2) / 3

Then the inverse relation of the given algebraic relation is, R^-1 = {(x, y): y = (x - 2) / 3}.

Try graphing y = 3x + 2 and y = (x - 2) / 3 and see whether the two graphs are symmetric about the line y = x.

Statement: For any relation R, (R^-1)^-1 = R.

Proof:

Here is the proof of the inverse relation theorem.

Let (x, y) ∈ R

⇔ (y, x) ∈ R^-1

⇔ (x, y) ∈ (R^-1)^-1

Thus, (R^-1)^-1 = R.

Hence proved.

Important Notes on Inverse Relation:

Here are some important points to note about inverse relationship.

• Domain and inverse of a relation are nothing but the range and domain of its inverse relation respectively.
• If R is a symmetric relation, R = R^-1.
• The inverse of an empty relation is itself. i.e., if R = { } then R^-1 = { }.
• On a graph, the curves corresponding to a relation and its inverse are symmetric about the line y = x.

Related Topics:

Explore the following related topics of inverse relation.

• Antisymmetric Relation
• Functions
• Transformation of functions
• Composite Functions
• Odd Functions

What Is Inverse of a Relation?

An inverse relation of a relation is a set of ordered pairs which are obtained by interchanging the first and second elements of the ordered pairs of the given relation. i.e., if R = {(x, y): x ∈ A and y ∈ B} then R^-1 = {(y, x): y ∈ B and x ∈ A}.

What Is Inverse Relation of an Empty relation?

If (x, y) is an element of a relation R, then (y, x) will be an element of its inverse relation R^-1. Thus, the inverse of an empty relation is itself. i.e., if R = { }, then R^-1 = { } as well.

What Is the Domain of an Inverse Relationship?

Let us consider a relation R and its inverse relation R^-1. Then the domain of R^-1 is the range of R.

How To Find the Inverse of an Algebraic Relation?

To find the inverse of an algebraic relation in terms of x and y, just interchange the variables x and y, and solve the equation for y. For example, to find the inverse of a relation y = x³, interchange x and y and then solve it for y. Then we get x = y³ ⇒ y = x^1/3.

What Is the Range of an Inverse Relation?

Let us consider a relation R and its inverse relation R^-1. Then the range of R^-1 is the domain of R.

What Is the Inverse Relationship of R If R Is Symmetric?

If R is symmetric, then (y, x) is in R for every (x, y) in R. Thus, its inverse relation is R itself. Note that whenever R = R^-1, then R is symmetric.

How To Find the Inverse of a Relation Given by a Graph?

To find the inverse relation of a graph, just draw the reflection of the graph along the line y = x. For this, we can pick some points on the graph, interchange their x and y coordinates to get the points on its inverse graph, plot the points and join them by a curve.