Problem 74 - Algebra 1

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

Step-by-Step Solution

Answer

a) The set of positive integers is not closed under subtraction. b) The set of integers is closed under addition and subtraction. c) The set of integers is closed under multiplication. d) The set of integers is not closed under division.

Step 1: Understanding the closure property

The closure property in mathematics states that given a set and an operation, if the operation's result always belongs to the given set for any two elements in the set, the set can be considered as 'closed' under the operation.

Step 2: Analyzing positive integers under subtraction

Consider any two positive integers, say, 5 and 3. The result of their subtraction, i.e., 5-3=2, is also a positive integer. So, it might seem that the set of positive integers is closed under subtraction. But what if we reverse the operation to 3-5? The result will not be a positive integer. Therefore, the set of positive integers is not closed under subtraction.

Step 3: Analyzing integers under addition and subtraction

The set of integers contains both positive and negative numbers. So, if we take any two integers and either add or subtract them, the result will always be another integer. Thus, the set of integers is closed under addition and subtraction.

Step 4: Analyzing integers under multiplication

If we multiply any two integers, positive or negative, the result will always be an integer. Thus, the set of integers is closed under multiplication.

Step 5: Analyzing integers under division

With division, if we take two integers, say 3 and 2, and divide 3 by 2, the result is 1.5 which is not an integer. Hence, the set of integers is not closed under division.

Key Concepts

Understanding Positive Integers

Positive integers, also known as natural or counting numbers, are the numbers we use for ordinary counting - 1, 2, 3, and so on. They do not include zero, negative numbers, fractions, or decimals.

When we perform operations on positive integers, the results can vary depending on the operation. For addition, the closure property holds true because if we add any two positive integers together, their sum will always be a positive integer. The concept is intuitive; imagine you have 3 apples and someone gives you 2 more, you now have 5 apples - still a positive number of apples.

However, subtraction with positive integers can be tricky. If you have a larger number and you subtract a smaller number from it, you're left with another positive integer. But, reverse that, subtract a larger number from a smaller one, and you enter the realm of negative numbers. Since negative numbers aren't part of the set of positive integers, this set lacks closure under subtraction.

Operations with Integers

The set of integers is much broader than just positive numbers; it includes negative numbers and zero. This set is represented by \( ... , -3, -2, -1, 0, 1, 2, 3, ... \). Integers operations include addition, subtraction, multiplication, and division.

When adding or subtracting two integers, the result stays within the realm of integers. For example, combining \( -2 \) and \( 3 \) through addition yields \( 1 \) and through subtraction yields \( -5 \), confirming the closure property for addition and subtraction.

Multiplication also adheres to the closure property. Multiplying any two integers will always produce another integer - multiplying \( -2 \) by \( 3 \) gives us \( -6 \), which is still an integer. It's helpful to remember that multiplying two negatives gives a positive result, while multiplying a negative with a positive results in a negative. However, division is where integers don't always play nice; dividing two integers won't necessarily result in another integer, as seen when dividing \( 3 \) by \( 2 \), getting a non-integer \( 1.5 \).

The Role of Mathematical Sets

In mathematics, a set is essentially a collection of distinct elements or numbers. Think of it as a basket where you can put numbers, and these numbers can follow certain rules or share common characteristics.

One key aspect of sets is that they can be defined to include or exclude certain numbers, such as the set of positive integers or the set of all integers. Depending on how a set is defined, the operations performed on the elements within that set can yield different manners of closure.

For example, the set of all integers remains closed under operations like addition, subtraction, and multiplication because no matter which two integers you pick from the set and perform these operations on, you will always end up with another integer - a member of the set. Such properties are vital in various fields of mathematics, including algebra and number theory, as they define the behavior of numbers under certain conditions.