Lecture Notes: Subsets of Real Numbers

Overview

• Discussion on specific subsets of real numbers
• Importance of decimal expansions in comparing real numbers
• Introduction to intervals as a way to construct subsets of real numbers

Decimal Expansion and Comparison

• Real numbers can be compared due to their decimal expansions.
• Complex numbers lack this comparison feature.
• Method of comparison: Look at the first point in the decimal expansion where numbers differ.
• Notable exception: Special cases involving repeated decimals (e.g., 0.999...)

Intervals

Closed Intervals

• Definition: If a and b are real numbers, the closed interval [a, b] includes all real numbers x such that a ≤ x ≤ b.
• Example: Intersection of closed intervals [0, 2] and [1, 3] results in [1, 2].
• Union: Union of [0, 2] and [1, 3] results in [0, 3].
• Symmetric Difference: Involves elements in one set but not both, e.g., (0, 1) and (2, 3) for symmetric difference.

Open Intervals

• Definition: Interval (a, b) includes real numbers x where a < x < b (endpoints not included).
• Example: Open interval (1, 2) includes 1.0001 but not 1.

Special Cases

• Empty Set: Occurs when b < a, resulting in an interval like [2, 0] or (2, 0).
• Single Element: [a, a] results in a set containing just {a}.
• Example: [2, 2] results in the set {2}.

Notation for Intervals Involving Infinity

• Minus Infinity: (-∞, a) includes all numbers less than a.
  • Can include a using (-∞, a].
• Plus Infinity: (b, +∞) includes all numbers greater than b.

Conclusion

• Understanding and using intervals is crucial for constructing subsets of real numbers.
• Intervals provide a systematic way to explore unions, intersections, and other operations on real number subsets.