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Chapter 5: Problem 3

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

Short Answer

Expert verified
{(-1, -1), (4, -3)}

Step by step solution

01

- Understand the Concept of Inverse Relation

The inverse of a relation is obtained by swapping the first and second elements in each ordered pair. If the pair is \(a, b\), its inverse will be \(b, a\).
02

- Apply the Definition to Each Pair

Given the relation: \(\{(-1, -1), (-3, 4)\}\), swap the elements in each pair.
03

- Swap Elements of Each Pair

Perform the swaps: \((-1, -1) \rightarrow (-1, -1)\) and \((-3, 4) \rightarrow (4, -3)\).
04

- Write the Inverse

After swapping, the inverse relation is \(\{(-1, -1), (4, -3)\}\).

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Key Concepts

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

swapping elements
To understand inverse relations, the key concept is swapping elements. When you have an ordered pair \(a, b\), you simply interchange the positions of \(a\) and \(b\). This means \(a\) moves to where \(b\) was and \(b\) moves to where \(a\) was. Think of it like changing seats. If \(a\) and \(b\) were on a see-saw, swapping elements means they switch places. This simple action of swapping is the core of finding inverse relations.
ordered pairs
An ordered pair is a fundamental concept in mathematics. It's referred to as 'ordered' because the sequence of elements is specific and significant. For example, in the pair \(-1, 4\), \(-1\) is in the first position and \(4\) is in the second position. Paying attention to this order is crucial when finding inverse relations. When we talk about swapping elements, we refer to the positions within the ordered pair. Thus, \(-1, 4\) swapped becomes \(4, -1\).
relation inversion
Relation inversion involves changing each ordered pair in a given relation into its inverse. Using our exercise, we start with the pairs \(-1, -1\) and \(-3, 4\). The inverse of \(-1, -1\) remains \(-1, -1\) since the elements are the same. For \(-3, 4\), swapping positions yields \(4, -3\). The final inverse relation is the collection of these new pairs, resulting in \{(-1, -1), (4, -3)\. This process emphasizes the simple yet important step of swapping elements within each pair to find their inverse.

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Solve using any method. $$5^{2 x}-3 \cdot 5^{x}+2=0$$
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