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Chapter 1: Problem 7

Prove using the notion of without loss of generality that \(\min (x, y)=(x+y-|x-y|) / 2\) and \(\max (x, y)=(x+y+\) \(|x-y| ) / 2\) whenever \(x\) and \(y\) are real numbers.

Short Answer

Expert verified
The formulas for min and max hold by considering both cases where x is greater than or less than y.

Step by step solution

01

- Define min and max

Let us start by defining the min and max functions for two real numbers x and y. \ \(\text{min}(x, y)\) gives the smaller value of x and y. \ \(\text{max}(x, y)\) gives the larger value of x and y.
02

- Consider two cases for \(x\) and \(y\)

Without loss of generality, assume two cases for x and y: \ Case 1: \(x \geq y\) \ Case 2: \(x < y\) \We'll prove that the given formulas hold for both cases.
03

- Case 1: \(x \geq y\)

If \(x \geq y\), then \ \( |x - y| = x - y \). \ Compute the min and max using the given formulas: \ \( \text{min}(x, y) = \frac{x + y - |x - y|}{2} = \frac{x + y - (x - y)}{2} = \frac{2y}{2} = y \), which is true because y is the smaller value. \ \( \text{max}(x, y) = \frac{x + y + |x - y|}{2} = \frac{x + y + (x - y)}{2} = \frac{2x}{2} = x \), which is true because x is the larger value.
04

- Case 2: \(x < y\)

If \(x < y\), then \ \( |x - y| = y - x \). \ Compute the min and max using the given formulas: \ \( \text{min}(x, y) = \frac{x + y - |x - y|}{2} = \frac{x + y - (y - x)}{2} = \frac{2x}{2} = x \), which is true because x is the smaller value. \ \( \text{max}(x, y) = \frac{x + y + |x - y|}{2} = \frac{x + y + (y - x)}{2} = \frac{2y}{2} = y \), which is true because y is the larger value.
05

- Conclusion

Since the given formulas hold true for both cases, we can conclude that \ \( \text{min}(x, y) = \frac{x + y - |x - y|}{2} \) and \ \( \text{max}(x, y) = \frac{x + y + |x - y|}{2} \) for any real numbers x and y.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Case Analysis
Case analysis involves breaking down a problem into simpler, mutually exclusive cases and solving each case independently. This method is particularly effective when dealing with expressions that change behavior under different conditions.
In our exercise, we used two cases:
  • Case 1: When x is greater than or equal to y
  • Case 2: When x is less than y
By analyzing these cases separately, we ensure that we account for all possible scenarios between the two real numbers x and y. This method helps simplify calculations and makes it easier to prove the given formulas for min and max functions.
In both scenarios, the absolute value term \(|x - y|\) will alter based on the relationship between x and y, influencing the resulting expressions for \(\text{min}(x, y)\) and \(\text{max}(x, y)\).
Absolute Value
The absolute value of a number is its distance from zero on the number line, regardless of direction. Mathematically, the absolute value of a number a is denoted as \(|a|\) and is defined as:
\[ |a| = \begin{cases} a & \text{if } a \geq 0 \ -a & \text{if } a < 0 \ \end{cases} \]
In our exercise, the absolute value is crucial because it determines whether the difference \(|x - y|\) will be positive or negative, hence impacting the min and max functions.
Consider our cases:
  • When x \geq y, \(|x - y|\) is simply x - y since the difference is non-negative.
  • When x < y, \(|x - y|\) becomes y - x because the difference is negative, and taking the absolute value flips the sign.
This behavior helps transform the general expressions for min and max into specific forms depending on the relative magnitudes of x and y.
Proof Techniques
Proof techniques are foundational tools in mathematics used to demonstrate the truth of statements. The proof technique used in our exercise is case analysis, where we systematically explored both possible scenarios of the relationship between x and y.
To structure our proof, we:
  • Defined the min and max functions.
  • Considered all possible relationships between x and y by analyzing two cases.
  • Applied the absolute value definition to simplify expressions.
  • Verified that the derived expressions matched the given formulas for each case.
Other common proof techniques include direct proof, proof by contradiction, and mathematical induction. Each has its use cases, but case analysis is particularly effective for problems like ours, where different conditions (i.e., x being larger or smaller than y) distinctly influence the outcome.
The logical flow in our proof ensures that no matter the values of x and y, the formulas for \(\text{min}(x, y)\) and \(\text{max}(x, y)\) hold true.

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