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Chapter 5: Problem 1

Find the inverse of the relation. $$\\{(7,8),(-2,8),(3,-4),(8,-8)\\}$$

Short Answer

Expert verified
{(8, 7), (8, -2), (-4, 3), (-8, 8)}

Step by step solution

01

Understand the Inverse of a Relation

The inverse of a relation is found by swapping each pair's elements. If the original relation is \(a, b\), then the inverse relation is \(b, a\).
02

Swap Each Pair

Take each pair from the original set and interchange the x and y values. (7, 8) becomes (8, 7) (-2, 8) becomes (8, -2) (3, -4) becomes (-4, 3) (8, -8) becomes (-8, 8)
03

Form the Inverse Relation

Collect all the swapped pairs to form the inverse relation: \(8, 7\), \(8, -2\), \(-4, 3\), \(-8, 8\). The inverse relation is therefore: \{(8, 7), (8, -2), (-4, 3), (-8, 8)\}

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Relation in Mathematics
A relation in mathematics describes a relationship between two sets of values. It is generally represented as a set of ordered pairs, where each pair includes an element from one set and an element from another set. For example, in the relation \(\{(7,8),(-2,8),(3,-4),(8,-8)\}\), each pair connects one value from the first set to one value from the second set.

It's essential to understand that a relation can establish a broad variety of connections between elements. Some examples include:
  • Mapping students to their grades
  • Assigning employees to projects
  • Pairing cities with their distances
The more you familiarize yourself with relations, the easier it will be to understand concepts like functions and inverse relations.
Inverse Functions
An inverse function involves reversing the roles of the input and output values in a relation. When we find an inverse, we swap the elements of each ordered pair. This means if we have a pair \(a, b\), the inverse would be \(b, a\).

Consider the initial relation \(\{(7,8),(-2,8),(3,-4),(8,-8)\}\). To find its inverse, we switch the x and y values:
  • (7, 8) becomes (8, 7)
  • (-2, 8) becomes (8, -2)
  • (3, -4) becomes (-4, 3)
  • (8, -8) becomes (-8, 8)
So, the inverse relation is \(\{(8, 7), (8, -2), (-4, 3), (-8, 8)\}\).

Finding inverses is particularly useful in various fields, like solving equations or transforming geometric objects.
Ordered Pairs
In mathematics, ordered pairs are fundamental to describing relations. An ordered pair is written as \(a, b\), where \(a\) is the first element (or x-value) and \(b\) is the second element (or y-value). The 'ordered' part means that \(a, b\) is not the same as \(b, a\).

Ordered pairs are vital in the context of functions, relations, and mappings. For example:
  • In the pair (3, 5), 3 is an element from the first set (often called the domain), and 5 is an element from the second set (called the range).
  • Swapping the components gives you a new ordered pair. The pair (3, 5) becomes (5, 3). This is a fundamental process in finding the inverse of a relation.
Understanding ordered pairs helps in visualizing relationships between variables. In our initial example, the ordered pairs were swapped to find the inverse relation, illustrating their importance in solving problems.

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Most popular questions from this chapter

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