91Ó°ÊÓ

The absolute value of a number, denoted as \(|a|\), is a fundamental concept in mathematics that refers to the distance of a number from zero on the number line. It's always a non-negative value.
Here's how it works:

• If the number is positive or zero, its absolute value is the number itself. For example, \(|5| = 5\), \(|0| = 0\).
• If the number is negative, its absolute value is the positive of that number. For instance, \(|-3| = 3\).

In the context of ordered integral domains, this is neatly defined as:
\[|a| = \begin{cases}a & \text{if} & a \geq 0 \-a & \text{if} & a < 0\end{cases}\]This ensures that regardless of positive or negative inputs, the absolute value represents the same non-negative magnitude of a number.

Inequalities are equations that describe the relative size or order of two values. They take various relational forms like '<', '>', '≤', and '≥'. In this exercise, we are dealing with the inequality \(|a| \leq b\).
This expression tells us that the absolute value of \(a\) should not exceed the positive number \(b\).Transforming absolute value inequalities can lead us to compound inequality statements:

• For \(|a| \leq b\), it translates to \(-b \leq a \leq b\) when considering both positive and negative scenarios.
• This translation splits the absolute value inequality into two separate comparisons: checking for \(a\) within the bounds of \(-b\) and \(b\).

These insights help in understanding how to manipulate and interpret inequalities involving absolute values logically in mathematical proofs.

A mathematical proof is a logical argument demonstrating the truth of a given statement. In this case, we're proving that \(|a| \leq b\) if and only if \(-b \leq a \leq b\) in an ordered integral domain.
This involves showing both directions:

Proving 'If' Direction

• Assume \(|a| \leq b\). This implies two possible scenarios:
• If \(a \geq 0\), then \(a \leq b\) follows directly from the definition.
• If \(a < 0\), then \(-a \leq b\), leading to \(a \geq -b\).

Thus, both cases combined give us \(-b \leq a \leq b\).

Proving 'Only If' Direction

• Assume \(-b \leq a \leq b\). We show this implies \(|a| \leq b\).
• If \(a \geq 0\), it directly leads to \(|a| = a \leq b\).
• If \(a < 0\), then from \(a \geq -b\), it ensures \(-a \leq b\), making \(|a| = -a \leq b\).

Both directions confirm the equivalence between \(|a| \leq b\) and \(-b \leq a \leq b\).

An Integral Domain is a type of ring with no divisors of zero where the multiplication is commutative, and it contains a multiplicative identity. Let's simplify these terms:

• Ring: A set accompanied by two operations, typically addition and multiplication, following specific rules.
• No Zero Divisors: In an integral domain, if the product of two numbers is zero, then at least one of the numbers must be zero.
• Commutative Multiplication: Ensures the order of multiplication doesn't alter the result, i.e., \(ab = ba\).
• Multiplicative Identity: There exists an element (usually 1) such that any number multiplied by it remains unchanged.

In the context of this exercise, an ordered integral domain further implies a structure where elements are not only numerical but have an assigned order reflecting their comparative size. This order aids in defining and understanding inequalities, thus playing a vital role in proving statements involving conditions like \(|a| \leq b\).