Lecture on Set Comprehension and Set Theory

Introduction to Set Comprehension

• Set comprehension is a method used to define subsets of numbers, particularly infinite ones.
• Example: Set of even integers defined by X mod 2 = 0.
  • X mod 2 = 0 means X is divisible by 2 without remainder, hence an even number.
• Set comprehension requires a starting set to construct a subset by applying a condition.

Constructing Subsets Using Set Comprehension

• Begin with a larger set (e.g., integers Z) and apply a condition to form subsets.
• Example notation: {X in Z | X mod 2 = 0} collects all even integers.

Examples of Set Comprehension

Perfect Squares

• An integer is a perfect square if its square root is also an integer.
• Constructing the set: {M in N | sqrt(M) in N} results in numbers like 1, 4, 9, 16, 25.

Rationals in Reduced Form

• A rational number is in reduced form if its numerator and denominator have no common divisors other than 1.
• Example: P/Q is in reduced form if GCD(P, Q) = 1.

Intervals of Numbers

• Defining intervals such as integers between -6 and 6.
• Syntax may vary but represents the same set.
• Real number intervals:
  • Closed Interval: Includes endpoints (e.g., [0, 1]).
  • Open Interval: Excludes endpoints (e.g., (0, 1)).
  • Mixed intervals (e.g., (0, 1]) are possible, named by which endpoints are included/excluded.

Operations on Sets

Union

• Combines elements from two sets into a single set without duplicates.
• Notation: X ∪ Y.
• Venn diagram represents overlapping elements in two sets.

Intersection

• Consists of elements common to both sets.
• Notation: X ∩ Y.

Set Difference

• Elements in one set and not in the other.
• Notation: X - Y or X \ Y.

Complement

• Elements not in the specified subset but within a universe set.
• Requires a specified universe to define what is excluded.

Conclusion

• Sets are fundamental in representing collections of mathematical objects.
• Understanding set operations and notations is crucial for mathematical discussions and applications.
• Both finite and infinite sets are significant in various contexts, from computational data to theoretical mathematics.