Problem 5 - Applied Discrete Structures

Question

Define \(a * b\) by \(|a-b|\), the absolute value of \(a-b\). Which properties does \(*\) have on the set of natural numbers, \(\mathbb{N} ?\)

Step-by-Step Solution

Answer

The operation \(*\) is commutative on \(\mathbb{N}\), but isn't associative and lacks identity and inverse elements.

Step 1: Understanding the Operation

The operation \( a * b \) is defined as the absolute value of the difference between two numbers, \( |a - b| \). This means that given any natural numbers \( a \) and \( b \), \( a * b \) will always yield a non-negative integer.

Step 2: Commutativity Check

To check if the operation is commutative, we need to verify if \( a * b = b * a \) for all \( a, b \in \mathbb{N} \). Since \( a * b = |a-b| \) and \( b * a = |b-a| \), and \( |a-b| = |b-a| \) by the properties of absolute value, the operation is commutative.

Step 3: Associativity Check

To determine if the operation is associative, investigate if \( (a * b) * c = a * (b * c) \). Calculate both expressions: \((a * b) * c = |(a * b) - c| \) and \(a * (b * c) = |a - (b * c)| \). Since the absolute value does not distribute over addition and subtraction in such a manner, the operation is not associative.

Step 4: Identity Element Check

For an identity element \( e \) in \( \, \mathbb{N} \), we need \( a * e = a \) for all \( a \). \( a * e = |a - e| = a \) implies that \( e = 0 \), but 0 is not a natural number (as the smallest natural number is 1), thus there is no identity element.

Step 5: Inverse Element Check

For each \( a \) there must exist some \( a' \) in \( \, \mathbb{N} \) such that \( a * a' = e \), where \( e \) is the identity, which we've previously reasoned does not exist in the set of natural numbers. Therefore, no inverse element exists for any \( a \).

Key Concepts

Commutative Operation

A commutative operation is one where the order of the operands does not affect the result. In simpler terms, if you switch the numbers around, you'll still get the same answer. For our operation defined as \( a * b = |a - b| \), it involves calculating the absolute value of the difference between \( a \) and \( b \). The absolute value function inherently satisfies the property of commutativity because \( |a - b| \) is equal to \( |b - a| \). This is due to the fact that the absolute value cancels out any negative signs and simply measures the distance between two numbers.

Students often remember this by thinking about how commuting to different locations doesn't change the distance traveled, just like swapping numbers around in an operation doesn't change the result.

• Commutativity is about swapping positions without altering the calculation.
• For example, \( 5 * 3 = |5 - 3| = 2 \) and \( 3 * 5 = |3 - 5| = 2 \).

If you can rearrange the operands and the outcome remains the same, you've got a commutative operation.

Associative Operation

An associative operation is one where the grouping of numbers does not impact the outcome. For example, when you're adding numbers, it doesn't matter if you add the first two together and then the third, or if you group the last two numbers first. Unfortunately, our defined operation \( a * b = |a - b| \) is not associative.

Let's investigate why: for associativity to hold true, \( (a * b) * c \) should equal \( a * (b * c) \). When you apply our operation, you find that \( (a * b) * c = |(a * b) - c| \) and \( a * (b * c) = |a - (b * c)| \). Since absolute values don't distribute over addition and subtraction in this context, these expressions are not equal.

• Lack of associativity means regrouping changes the result.
• If expressions are not equal, the operation can't be associative.

It's important to understand that associativity failing indicates the operation doesn't handle multiple components evenly like addition or multiplication do.

Identity Element

The identity element in an operation is a special number that doesn't change other numbers when used in the operation. Imagine adding zero to a number, the number remains the same, therefore zero is the identity element for addition. However, for our operation \( a * b = |a - b| \), we encountered a challenge finding a natural number that acts as an identity element.

To determine the identity element, we need \( a * e = a \) for all natural numbers \( a \). This simplifies to \( |a - e| = a \), implying \( e \) should be zero for the operation to leave \( a \) unchanged. Unfortunately, zero is not part of the natural numbers, which typically start at 1, so no identity element exists in this context.

• An identity element should not alter other elements.
• Absence of zero in natural numbers means no identity element here.

Understanding the role of an identity element helps to identify when one is missing, which can significantly affect the structure of the set and its operations.

Inverse Element

An inverse element in an operation pairs with another element to yield the identity element. It's like adding a number and its negative to get zero in addition. In our specific operation \( a * b = |a - b| \), we find ourselves searching for an inverse element but run into trouble.

For each \( a \), there should exist an \( a' \) such that \( a * a' = e \), where \( e \) is the identity element. The finding earlier was that there was no valid identity element within natural numbers, as it would be zero, not satisfying the natural number criteria. Without an identity element, we can't define what an inverse element should be, making it impossible for an inverse to exist in this setting.

• Inverse elements rely on the presence of an identity element.
• Without zero, we cannot establish an identity or inverse in natural numbers.

Recognizing the interdependence of identity and inverse elements emphasizes the significance of entire algebraic structures where each element interconnects.