Problem 12 - Algebra 2

Question

Find the inverse of each relation. $$ \\{(7,-4),(3,5),(-1,4),(7,5)\\} $$

Step-by-Step Solution

Answer

The inverse relation is \( \{(-4, 7), (5, 3), (4, -1), (5, 7)\} \).

Step 1: Understand the Concept of Inverse

The inverse of a relation is formed by swapping the components of each ordered pair in the relation.

Step 2: Swap Each Ordered Pair

To find the inverse of the relation \( \{(7,-4),(3,5),(-1,4),(7,5)\} \), we swap the components of each pair:- Pair \((7, -4)\) becomes \((-4, 7)\)- Pair \((3, 5)\) becomes \((5, 3)\)- Pair \((-1, 4)\) becomes \((4, -1)\)- Pair \((7, 5)\) becomes \((5, 7)\)

Step 3: Write the Inverse Relation

After swapping each ordered pair, the inverse relation becomes: \[ \{(-4, 7), (5, 3), (4, -1), (5, 7)\} \].

Key Concepts

Understanding Ordered Pairs

An ordered pair is a fundamental concept in mathematics, especially in the study of relations. It is written in the form \((x, y)\), where \(x\) is the first component and \(y\) is the second component. Ordered pairs are the building blocks of more complex mathematical structures like relations and functions. They help in describing the relationship between two quantities. For example, the ordered pair \((7, -4)\) indicates that 7 is related to -4. The position of each number is crucial because it dictates the nature of the relation. Changing the order of the components yields a different pair. This is where the concept of ordered pairs intersects with the idea of inverse relations. These pairs can represent anything from coordinates in geometry to values in algebraic expressions. They are usually enclosed within braces \(\{ \} \) to form a set, helping to display all connections within the relation comprehensively.

Exploring Relations in Mathematics

A relation in mathematics is essentially a set of ordered pairs. It establishes a connection between elements of two sets. For instance, the relation \( \{(7, -4), (3, 5), (-1, 4), (7, 5)\} \) includes multiple connections between different values of \(x\) and \(y\). In mathematical terms, if we have a set \(A\) comprising the first elements of each ordered pair and a set \(B\) comprising the second elements, the relation describes how elements of set \(A\) are associated with elements of set \(B\). Relations are integral to mathematics because they lay the groundwork for types like functions, wherein every input corresponds to exactly one output. It's important to understand the properties of relations such as reflexivity, symmetry, and transitivity, particularly when dealing with more sophisticated mathematical concepts. In our relation example, determining the inverse essentially means swapping these connections, which leads us to our next concept.

Swapping Components: Finding the Inverse

Swapping components is the key action in finding the inverse of a relation. This involves exchanging the positions of the first and second elements in each ordered pair. This simple yet powerful technique creates a new set of ordered pairs, referred to as the inverse relation.To illustrate, consider our original relation \( \{(7, -4), (3, 5), (-1, 4), (7, 5)\} \). By swapping their components, we form a new relation:

• Pair \((7, -4)\) becomes \((-4, 7)\).
• Pair \((3, 5)\) becomes \((5, 3)\).
• Pair \((-1, 4)\) becomes \((4, -1)\).
• Pair \((7, 5)\) becomes \((5, 7)\).

The resulting inverse is \( \{(-4, 7), (5, 3), (4, -1), (5, 7)\} \). This inversing technique is crucial in advanced mathematics, particularly in solving equations and understanding functions and graphs. It is a valuable skill for students to master as it often appears in various mathematical problems.