Beyond the Finite
Sometimes sets don't have bounds. To fix this, we extend the real number system by adding two new symbols: \( +\infty \) and \( -\infty \). The Extended Real Number System is denoted as \( \overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\} \).
Warning: \( \pm\infty \) are NOT real numbers. You cannot do standard arithmetic with them (e.g., \( \infty - \infty \) is undefined). They are primarily used for ordering and describing unbounded sets.
Interactive: The Extended Line
Select a set to see its supremum and infimum in the extended system. Notice how unbounded sets now have a defined sup/inf!
Arithmetic with Infinity
Allowed Operations
- \( a + \infty = \infty \) (for \( a \neq -\infty \))
- \( a - \infty = -\infty \) (for \( a \neq \infty \))
- \( a \cdot \infty = \infty \) (if \( a \gt 0 \))
- \( a \cdot \infty = -\infty \) (if \( a \lt 0 \))
- \( \frac{a}{\infty} = 0 \) (for \( a \in \mathbb{R} \))
Undefined Operations
- \( \infty - \infty \)
- \( 0 \cdot \infty \)
- \( \frac{\infty}{\infty} \)
- \( \frac{a}{0} \) (still undefined)
Practice Problems
True/False
1. The symbol \( +\infty \) is a real number.
2. For any non-empty set \( S \subseteq \mathbb{R} \), \( \sup S \) always exists in the extended real number system.
3. If \( \sup S = +\infty \), then the set \( S \) has a maximum element.
4. The expression \( +\infty + (-\infty) \) is defined and equals 0.
Fill in the Blank
1. The extended real number system is denoted by \( \overline{\mathbb{R}} = \mathbb{R} \cup \{ \_\_\_\_\_ \} \).
2. If a set \( S \) is not bounded above, then \( \sup S = \_\_\_\_\_ \).
3. The interval \( (-\infty, 5] \) represents the set \( \{x \in \mathbb{R} : x \_\_\_\_\_ 5\} \).
4. If \( A = [1, \infty) \) and \( B = [2, \infty) \), then \( \sup(A+B) = \_\_\_\_\_ \).
Full Problems
1. Let \( S = \{x \in \mathbb{R} : x^2 \gt 4\} \). Find \( \sup S \) and \( \inf S \) in the extended real number system.
First, solve the inequality \( x^2 \gt 4 \). This means \( x \gt 2 \) or \( x \lt -2 \).
So \( S = (-\infty, -2) \cup (2, \infty) \).
Since \( S \) contains arbitrarily large positive numbers, it is not bounded above. Thus, \( \sup S = +\infty \).
Since \( S \) contains arbitrarily large negative numbers (large magnitude), it is not bounded below. Thus, \( \inf S = -\infty \).
2. Let \( A = \{n^2 : n \in \mathbb{N}\} \). Find \( \sup A \) and \( \inf A \).
The set is \( A = \{1, 4, 9, 16, \dots\} \).
The smallest element is 1, so \( \inf A = 1 \).
The set is not bounded above because \( n^2 \) grows without bound as \( n \) increases. Thus, \( \sup A = +\infty \).
3. Let \( C = \{x \in \mathbb{Q} : x \lt 0\} \). Find \( \sup C \) and \( \inf C \).
The set consists of all negative rational numbers.
It is bounded above by 0, and there are elements arbitrarily close to 0 (e.g., \( -1/n \)). Thus, \( \sup C = 0 \).
It is not bounded below, as it contains arbitrarily large negative numbers (e.g., \( -n \)). Thus, \( \inf C = -\infty \).
4. Consider the set \( D = \mathbb{Z} \). Find \( \sup D \) and \( \inf D \).
The set of integers extends infinitely in both positive and negative directions.
Therefore, \( \sup D = +\infty \) and \( \inf D = -\infty \).
5. Let \( E = \{x \in \mathbb{R} : x^3 \lt 8\} \). Find \( \sup E \) and \( \inf E \).
The inequality \( x^3 \lt 8 \) is equivalent to \( x \lt 2 \).
So \( E = (-\infty, 2) \).
The set is bounded above by 2, and 2 is the least upper bound. Thus, \( \sup E = 2 \).
The set is not bounded below. Thus, \( \inf E = -\infty \).