Let X = {1, 2, 3, 4} and R = {(x, y) | x + y < 3}. Find the inverse of R and also give its matrix.

Introduction

In the field of discrete structures, a relation is a set of ordered pairs. The inverse of a relation R, denoted as R^-1, is simply the relation that results from swapping the elements in each ordered pair in R.

In this problem, we are given a set X = {1, 2, 3, 4} and a relation R = {(x, y) | x + y < 3}. This means that R contains all ordered pairs (x, y) such that the sum of x and y is less than 3.

Finding the Inverse of R

To find the inverse of R, we need to swap the elements in each ordered pair in R. The relation R is defined as R = {(x, y) | x + y < 3}. The pairs that satisfy this condition are (1,1) and (1,2). Therefore, R = {(1,1), (1,2)}.

The inverse of R, denoted as R^-1, is then obtained by swapping the elements in each ordered pair in R. This gives us R^-1 = {(1,1), (2,1)}.

Representing the Inverse of R as a Matrix

A relation matrix is a binary matrix that represents a relation. The rows and columns of the matrix correspond to the elements of the set, and the elements of the matrix are 1 if the corresponding ordered pair is in the relation, and 0 otherwise.

To create a relation matrix for the inverse of R, we start with a 4x4 matrix filled with zeros, since the set X has 4 elements. For each ordered pair in R^-1, we change the corresponding element in the matrix to 1.

This gives us the following matrix for R^-1:

Conclusion

In conclusion, the inverse of the given relation R = {(x, y) | x + y < 3} is R^-1 = {(1,1), (2,1)}, and its matrix representation is the 4x4 matrix shown above. These results are significant in the field of discrete structures, as they provide a way to manipulate and represent relations in a form that is easy to work with.

Examples

For further clarification, let's consider another example. Suppose we have a set Y = {1, 2, 3} and a relation S = {(x, y) | x - y = 1}. The pairs that satisfy this condition are (2,1) and (3,2), so S = {(2,1), (3,2)}. The inverse of S is then S^-1 = {(1,2), (2,3)}, and its matrix representation is:

Summary

The inverse of a relation R is obtained by swapping the elements in each ordered pair in R. The inverse of the given relation R = {(x, y) | x + y < 3} is R^-1 = {(1,1), (2,1)}. The matrix representation of R^-1 is a 4x4 matrix with the elements (1,1) and (2,1) set to 1.

Analogy

Imagine you have a set of friends and a relation that represents whether two friends are neighbors. The inverse of this relation would be a relation that represents whether two friends are not neighbors.