Inverse Relations And Functions Practice

In Set up theory, a relation R from A to B, where A and B are non-empty sets, is a subset of the Cartesian production of A and B, i.e. A × B. Here, the set of all first elements of the ordered pairs of R is called the domain of the relation, whereas the second element fix is called the range of the relation R. Besides, the unabridged gear up B is referred to every bit the codomain of R, and we know that range is always a subset of the codomain. In maths, we can meet different types of relations. In this article, you will acquire about inverse relations using graphical representation and examples.

Inverse Relation Definition

The inverse of a relation is a relation obtained by interchanging or swapping the elements or coordinates of each ordered pair in the relation. Inverse relation in sets can exist divers using the ordered pairs. The domain and range of an inverse relation tin can exist written by swapping the domain and range of that relation. That means the domain of relation will be the range of its inverse, and the range of relation volition exist the domain of its inverse.

Acquire more than about the domain and range of relations hither.

Mathematical Definition of Inverse Relation:

Suppose R is a relation of the course {(10, y): x ∈ A and y ∈ B} such that the inverse relation of R is denoted past R^-1 and R^-1 = {(y, x): y ∈ B and 10 ∈ A}. If R is from A to B, and so R^-ane is from B to A. In other words, if (x, y) ∈ R, so (y, 10) ∈ R^-ane and vice versa. Also, we know that relation in sets is a subset of the Cartesian product of sets, i.due east. R is a subset of A ten B, and R^-1 is a subset of B x A.

Inverse Relation Graph

If an equation describes the relation in the variables x and y, the equation of the changed relation is obtained by replacing every 10 in the equation with y and every y in the equation with x. Their graphs are mirror images over the line of reflection y = ten. Consider an ordered pair (a, b) of a relation expressed in terms of an equation in x and y, such that the inverse relation of this equation contains an ordered pair as (b, a) nigh the reflection on y = ten as shown in the beneath figure.

Inverse relation

Let's understand more about inverse relation with an example below.

Inverse Relation with Example

Question:

Find the changed of a relation R represented by {(-15, -4), (-18, -8), (-half dozen, 2), (-12.55, three)} and write the domain and range of the inverse relation.

Solution:

Given,

R = {(-15, -4), (-18, -8), (-6, ii), (-12.55, 3)}

Domain = {-fifteen, -18, -half dozen, -12.55}

Range = {-4, -viii, 2, 3}

Changed of R = R^-1 = {(-four, -15), (-8, -18), (2, -6), (3, -12.55)}

Domain of R^-i = {-4, -eight, two, three}

Range of R^-ane = {-15, -18, -half-dozen, -12.55}