Problem 29 - Book of Proof

Question

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) If \(x, y \in \mathbb{R}\) and \(|x+y|=|x-y|\), then \(y=0\).

Step-by-Step Solution

Answer

The statement is true. Whenever \( |x+y| = |x-y| \), it necessarily follows that \( y = 0 \).

Step 1: Rewrite the Equality

The given expression can be rewritten to separate the two scenarios covered by the absolute value: \n(i) \( x + y = x - y \) , which happens when \( x \geq y \), or \n(ii) \( x + y = -(x - y) \), which happens when \( x < y \).

Step 2: Solve for Scenarios

Solve for the two scenarios separately. \n(i) For the first scenario \( x + y = x - y \), simplifying this gives \( 2y = 0 \), implying \( y = 0 \). \n(ii) For the second scenario \( x + y = -(x - y) \), simplifying this gives \( 2y = 0 \) again, implying \( y = 0 \). Therefore, no matter which scenario, \( y = 0 \).

Step 3: Conclusion

From the solutions for the two scenarios, in both cases found that \( y = 0 \). Therefore, the original statement is true, i.e., when \( |x+y| = |x-y| \), then \( y = 0 \).

Key Concepts

Absolute Value

Understanding the concept of absolute value is fundamental in various fields of mathematics, including algebra and analysis. Essentially, the absolute value of a real number is its distance from zero on the number line, regardless of direction. In other words, it tells us how 'large' a number is without considering whether it's positive or negative.

Mathematically, the absolute value of a number x, denoted by |x|, is defined by:

• |x| = x, if x is non-negative (x is positive or zero)
• |x| = -x, if x is negative

When dealing with absolute values, one must consider both scenarios: the number being positive or negative. This dual nature leads to the necessity of breaking down equations into different cases, as seen in the exercise solution. The insight that |x+y| could equal |x-y| only under specific conditions is crucial in unraveling the problem given.

Real Numbers

The set of real numbers, denoted by \(\mathbb{R}\), encompasses all numbers that can be found on the number line, including both rational numbers (like fractions and integers) and irrational numbers (like \(\sqrt{2}\) and \(\pi\)). In the context of the given exercise, the variable x and y are members of this extensive set, meaning they could be any number from this continuous range.

Diving into the world of real numbers introduces the concept of operations on this set, including addition and subtraction as are present in the exercise. The properties of real numbers are applied during the problem-solving process, which include the assumption of variables behaving according to known arithmetic rules, such as distributive and commutative laws. These properties are implicitly used when the exercise splits the absolute value equation into two distinct cases.

Proof Techniques

Mathematical proofs are rigorous arguments that establish the truth of mathematical statements. Several proof techniques exist, including direct proof, proof by contradiction, and proof by induction, among others. In the exercise provided, a direct proof approach is taken, where a statement is shown to be true by direct reasoning.

To tackle the exercise, the proof begins by considering the implications of the given absolute value equation under different conditions. This method of breaking the problem down into different cases is itself a fundamental proof technique, especially when dealing with absolute values. It's essential for students to understand that transforming the statement step by step and deductively inferring the only possible solution ensures that the conclusion reached is logical and sound.

By expressing each step carefully and understanding how it moves the proof forward, students can comprehend the broader applications of proof techniques. Assigning separate cases for when x is greater than or equal to y, and when x is less than y, leads to a single conclusion that validates the initial claim, demonstrating a solid understanding of direct proofs.