Problem 2

Question

In any ordered integral domain, define $|a|$ by $$ |a|=\left\\{\begin{array}{rll} a & \text { if } & a \geq 0 \\ -a & \text { if } & a<0 \end{array}\right. $$ Using this definition, prove the following: $$ a \leq|a| $$

Step-by-Step Solution

Verified

Answer

For all values, $a \leq |a|$ by definition: when positive, $a=|a|$; when negative, $a\leq-a=|a|$.

1Step 1: Understanding the Absolute Value Definition

The exercise provides the definition of the absolute value function for any element $a$ in an ordered integral domain: $|a|$ is $a$ if $a\geq0$, and $-a$ if $a<0$. This definition will be used to prove that $a \leq |a|$.

2Step 2: Case 1: Non-Negative $a$

Consider the case where $a \geq 0$. According to the definition, $|a| = a$. With the inequality $a \leq |a|$, replacing $|a|$ with $a$ results in $a \leq a$, which is always true.

3Step 3: Case 2: Negative $a$

Now consider the case where $a < 0$. Here, $|a| = -a$. We need to show that $a \leq -a$. Since $a < 0$, it follows that $-a > 0$. Clearly, $a$, being negative, is less than any positive number, hence $a \leq -a$.

4Step 4: Combine Case Analyses

With both cases analyzed — when $a \geq 0$ and $a < 0$ — the statement $a \leq |a|$ holds true in both scenarios, thereby proving the equation for any element $a$ in the domain.

Key Concepts

Absolute Value in Ordered Integral DomainsProof of the Inequality $a \leq |a|$Utilizing Case Analysis in Problem Solving

Absolute Value in Ordered Integral Domains

In an ordered integral domain, understanding the concept of absolute value is essential. The absolute value of a number, denoted as $|a|$, represents the non-negative magnitude of that number. It is defined as follows: if $a$ is greater than or equal to zero, then the absolute value $|a|$ is simply $a$ itself. However, if $a$ is less than zero, its absolute value becomes the opposite $-a$, effectively making it positive.

This definition ensures that absolute values are always non-negative:

For $a \geq 0$: $|a| = a$
For $a < 0$: $|a| = -a$

The absolute value provides a way to measure the size or magnitude of elements without regard to their sign. This characteristic is crucial in establishing inequalities where only the magnitude is considered, such as in the next sections.

Proof of the Inequality $a \leq |a|$

One fundamental property of absolute value is the inequality $a \leq |a|$. This can be proven using the definition of absolute value and considering two separate cases based on the sign of $a$. Let's dive in.

**Case 1: $a \geq 0$**In this scenario, since $a$ is non-negative, the absolute value definition gives us $|a| = a$. The inequality $a \leq |a|$ simplifies to $a \leq a$, which is trivially true. Every number is equal to itself, ensuring this part of the proof is valid.

**Case 2: $a < 0$**Here, $a$ is negative, so by definition, $|a| = -a$, turning $a$ into a positive quantity. Given $a < 0$, we naturally have $-a > 0$. For any negative number $a$, the inequality $a \leq -a$ holds, since a negative number $a$ is always less than any positive number such as $-a$.

Combining these two cases confirms the inequality $a \leq |a|$ is universally true for any element $a$ in an ordered integral domain.

Utilizing Case Analysis in Problem Solving

Case analysis is a powerful method of proof, especially when dealing with definitions that depend on conditions, such as the absolute value function. By splitting a problem into distinct cases, we can address each condition independently. Here, our approach involved considering $a$ being non-negative in one instance, and negative in another.

Specific Conditions: Define scenarios based on meaningful conditions, e.g., $a \geq 0$ and $a < 0$.
Independent Proofs: Prove your statement independently within each case.
Combine Results: Ensure that every possible scenario is accounted for by using conclusions from all individual cases.

This technique is particularly helpful in mathematical proofs where properties can change depending on the input conditions. In this exercise, we successfully used case analysis to demonstrate the validity of the inequality $a \leq |a|$, confirming our understanding of absolute value and inequalities in ordered integral domains.

Problem 2