Question

A set of numbers is closed under an operation if applying the operation to any two numbers in the set results in another number in the set. For instance, positive integers are closed under addition because the sum of any two positive integers is a positive integer. Decide whether the set is closed under the given operation. a. positive integers; subtraction b. integers; addition and subtraction c. integers; multiplication d. integers; division

Name: Problem 74, Chapter 2: Algebra 1
Uploaded: 2020-08-31T00:04:19-08:00
Duration: 2 min 16 s
Channel: Amy Jiang

   A set of numbers is closed under an operation if applying the operation to any two numbers in the set results in another number in the set. For instance, positive integers are closed under addition because the sum of any two positive integers is a positive integer. Decide whether the set is closed under the given operation.
a. positive integers; subtraction
b. integers; addition and subtraction
c. integers; multiplication
d. integers; division

Algebra 1

Ron Larson, Laurie… 1st Edition

Chapter 2, Problem 74

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Solved by Expert Amy Jiang

08/31/2020

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We need to determine if the given sets are closed under the given operations. Show more…

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A set of numbers is closed under an operation if applying the operation to any two numbers in the set results in another number in the set. For instance, positive integers are closed under addition because the sum of any two positive integers is a positive integer. Decide whether the set is closed under the given operation. a. positive integers; subtraction b. integers; addition and subtraction c. integers; multiplication d. integers; division

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Key Concepts

Closure Property

The closure property is a fundamental concept in algebra which means that when an operation is applied to any two elements of a set, the result is still an element of that set. This property is essential for defining structures such as groups, rings, and fields where operations must remain within the set.

Binary Operations

Binary operations are calculations that combine two elements (for example, addition, subtraction, multiplication, and division) to produce another element. The nature of these operations, whether they preserve closure, depends on both the operation and the properties of the set, making this concept central in abstract algebra.

Set Theory

Set theory provides the underlying framework for understanding and analyzing collections of elements and their properties, including closure. It involves studying how operations manipulate elements within a set and determining whether such manipulations keep the results in the original set.

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