Problem 111 - Elementary and Intermediate Algebra

Question

A set is said to be closed under addition if the sum of any two of its members is also a member of the set. Is the set \(\\{-1,0,1\\}\) a closed set under addition? Explain.

Step-by-Step Solution

Answer

The set is not closed under addition because the sums \(-2\) and \(2\) are not in the set.

Step 1: Understand the Problem

We need to determine if the set \(\{-1, 0, 1\}\) is closed under the operation of addition. This means checking if the sum of any two numbers in the set results in a number that is also in the set.

Step 2: List All Possible Sums

Identify all pairs of elements that can be summed. The elements of the set are \(-1, 0,\) and \(1\), so we should check these pairs: \((-1, -1), (-1, 0), (-1, 1), (0, 0), (0, 1), (1, 1)\).

Step 3: Calculate Each Sum

Calculate the sum for each pair:- \((-1) + (-1) = -2\)- \((-1) + 0 = -1\)- \((-1) + 1 = 0\)- \(0 + 0 = 0\)- \(0 + 1 = 1\)- \(1 + 1 = 2\)

Step 4: Check if Sums are in the Set

Verify if each calculated result from Step 3 is an element of the set \(\{-1, 0, 1\}\):- \(-2\) is not in the set.- \(-1, 0, 1\) are in the set.- \(2\) is not in the set.

Key Concepts

Understanding Set Theory and Closure

Set theory is a branch of mathematical logic that studies collections of objects, known as sets. In the context of closure, we consider whether operations performed on elements within a set keep the results within the same set.

For example, when examining closure under addition, we ask if summing any two numbers within the set results in an outcome that is also a member of that set. This is a vital concept because it helps define the boundaries and properties of mathematical systems.

In our exercise, we are dealing with a set composed of the numbers \(-1, 0, 1\). Closure under addition would mean that adding any two numbers within this set results in a number that also belongs to the set. If every possible sum of the set's elements is also an element of the set, then the set is closed under the addition operation.

Addition in Algebra: Pairing and Summation

In algebra, addition is one of the fundamental operations. It involves two numbers and combines them into one sum. To explore closure in sets, we look at all possible pairs formed by the numbers in the set.

For the set \(-1, 0, 1\), examining every pairing means you make sure not to miss any possible combination, including duplicates such as \(-1\)+\(-1\) and pairs with self like \(0\)+\(0\). These calculations tell us whether every possible addition sticks to the criteria of closure.

• Pairs like \((-1, -1)\) lead to new numbers like \(-2\).
• Zero paired with any number gives the other number.
• Positive and negative numbers sum depending on their absolute values.

The key takeaway is reviewing each sum to see if it falls within the original set.

Analyzing Mathematical Operations for Closure

Mathematical operations help us understand how elements within a set interact under specified conditions. The focus here is addition, which combines two values into one new value. For a set to be 'closed' under addition, every resulting sum of its elements must remain within the set.

When assessing closure, you're essentially asking if the operation allows you to "stay" within the boundaries of the set—a bit like ensuring all discussions remain "in-house."

• Checking sums involving negative, positive, and zero provides the full picture.
• If just one result falls outside the set, then closure isn't achieved.

Given the numbers \(-2\) and \(2\) appeared as sums in the exercise but aren't part of the set, it demonstrates this set isn't closed under addition. This crucial analysis signifies a lack of closure because not all sums stayed inside the set's parameters.