Problem 26 - Discrete Mathematics with Applications

Question

Prove by cases, where \(n\) is an arbitrary integer and \(|x|\) denotes the absolute value of \(x\). $$|-x|=|x|$$

Step-by-Step Solution

Answer

We can prove that \(|-x| = |x|\) for an arbitrary integer x by considering three cases: 1. If x > 0, then \(|-x| = |-1\cdot x| = |x|\) 2. If x < 0, let y > 0 such that x = -y. Then, \(|-x| = |-(-y)| = |y| = |x|\) 3. If x = 0, then \(|-x| = |0| = 0 = |x|\) In all three cases, the statement holds true, proving \(|-x| = |x|\) for all integer values of x.

Step 1: Case 1: x > 0 (x is positive)

In this case, the number x is a positive integer. By definition, the absolute value of a positive number is the number itself. Therefore, we have: $$|-x|=|-1\cdot x|=|-1|\cdot|x| =1\cdot|x|=|x|$$

Step 2: Case 2: x < 0 (x is negative)

In this case, the number x is a negative integer. Notice that the absolute value of a negative integer is its opposite, which is positive. So, let's write x as the product of -1 and a positive number y (x = -y, where y > 0). Then we have: $$|-x|=|-(-y)|=|y|=|x|$$ since the absolute value of x is y in this case.

Step 3: Case 3: x = 0 (x is zero)

In this case, the number x is zero. The absolute value of zero is zero, so we have: $$|-x|=|-0|=0=|x|$$ In all three cases, the statement holds true, and we have proven that \(|-x|=|x|\) for all integers x.

Key Concepts

Proof by Cases

Understanding the concept of proof by cases enables us to tackle complex mathematical statements by breaking them into more manageable pieces. When a statement can occur in different scenarios, we can apply proof by cases to verify the truth of the statement in every individual circumstance, ensuring no possibilities are overlooked.

Consider a statement that depends on a variable that could belong to several distinct and non-overlapping cases. By splitting the proof into cases, you analyze each case separately, using logical steps to show the statement holds true. Once all possible cases are verified, collectively they establish the full proof of the original statement. Proving by cases is a strategy akin to covering all your bases in baseball—every scenario must be accounted for to claim victory.

Absolute Value

The absolute value of a real number is a measure of its magnitude regardless of its sign. In simpler terms, it tells us how far away the number is from zero on the number line. Symbolically, the absolute value of any number x is denoted as |x|.

Here's a fundamental property: the absolute value of any number and its negative is always the same. Hence, for every x, |x| is equal to |-x|. For instance, |-3|=3 and |3|=3; discovering that both expressions evaluate to the same number highlights the crucial concept that absolute value disregards the sign.

Integer Properties

Integers, the set of whole numbers and their negatives, embody a list of properties that provide a foundation for various branches of mathematics. Among these are the commutative, associative, and distributive properties, which govern how we can add, subtract, and multiply integers.

When we talk about absolute values in relation to integers, it's useful to recall these properties, particularly the distributive property. For absolute values, this property assures that multiplying a number by -1 and then finding its absolute value doesn't change the result since |-1 * x| = |-1| * |x| = |x|. Understanding such integer properties helps simplify complex expressions and perform proofs, like the absolute value proof discussed in this exercise.

Mathematical Proofs

A mathematical proof is a logical argument that establishes the truth of a mathematical statement beyond any doubt. Each proof is a sequence of deductive logical steps, grounded in axioms, definitions, and previously established theorems, culminating in the statement being proven.

In our exercise, the statement |-x| = |x| for all integers x needed a solid proof. By breaking it down into cases and analyzing each one separately with logical reasoning, we provided a mathematical proof ensuring the statement's veracity for all possible integer values of x. Mastery of such proofs is pivotal for mathematicians and students alike to navigate the realm of mathematics with confidence and precision.