Problem 4 - A Primer of Real Analysis

Question

Show that if \(u \in \mathbb{R},\) then one and only one of the following is true: (a) \(u>0,\) (b) \(u<0,\) or (c) \(u=0\)

Step-by-Step Solution

Answer

For any real number \(u\), one and only one of: \(u > 0\), \(u < 0\), or \(u = 0\) is true.

Step 1: Comprehend the Problem

The task is to prove that for any real number \(u\), only one of three possible conditions can be true at a time: \(u > 0\), \(u < 0\), or \(u = 0\). This must be shown for all real numbers \(u\) in \(\mathbb{R}\).

Step 2: Use the Definitions of Real Numbers

Understand that real numbers \(\mathbb{R}\) encompass all numbers that can be found on the number line, including positive numbers, negative numbers, and zero. A real number \(u\) must satisfy one of these conditions: it either lies to the right of zero (\(u > 0\)), to the left of zero (\(u < 0\)), or is exactly at zero (\(u = 0\)).

Step 3: Contradict the Possibility of Multiple Conditions Being True

Assume, for contradiction, that more than one of the conditions could be true for a real number \(u\). If \(u > 0\) and \(u < 0\) were both true, then \(u\) would be simultaneously greater and less than zero, which is impossible. Similarly, \(u\) cannot be zero and also be greater or less than zero at the same time.

Step 4: Prove Exclusivity of Each Condition

For any \(u > 0\), it cannot be true that \(u < 0\) or \(u = 0\) because being positive means it is greater than zero. Similarly, if \(u < 0\), \(u\) cannot be zero or positive. Lastly, if \(u = 0\), it is neither greater nor less than zero.

Step 5: Conclusion on Exclusivity and Exhaustiveness

We have shown that there are no overlaps between the three conditions and that every real \(u\) must satisfy exactly one of them. Every real number is either positive, negative, or zero, and exactly one of these properties applies.

Key Concepts

Positive Numbers

Positive numbers include all real numbers that are greater than zero. These numbers can be found to the right of zero on the number line. They simply represent values that are above zero and could include numbers like 1, 2.5, 100, and so on.

• They are always greater than zero.
• Examples: 3, 6.7, 1000.
• They have a positive sign (+) but it is usually omitted.

When working with positive numbers, it's important to remember that they signify amounts, distances, or quantities that add to the basic unit of "zero." For example, if you move 5 meters forward, you've increased your position by a positive number.

Negative Numbers

Negative numbers are numbers less than zero, found to the left of zero on the number line. These numbers represent values that 'subtract' from zero and appear with a minus (-) sign.

• They are always less than zero.
• Examples: -2, -5.5, -100.
• The negative sign indicates how far and in what direction a number is from zero.

Negative numbers are often used in contexts like temperature, money (debt), or elevations below sea level. They show that there is a deficit or a decrease from a neutral or zero point.

Zero

Zero is a unique number that represents neither positivity nor negativity. It plays a central role in the number line as the point that divides positive numbers and negative numbers.

• Zero is neither positive nor negative.
• It serves as the fundamental starting point for measuring distances, scores, and changes.
• It is the only number that is neither greater than nor less than itself.

The concept of zero acts as a reference or a baseline in mathematics and beyond. It is crucial in calculations, balances out equations, and is often used to signify nothing or no amount.