Understanding the Theory of Sets
1. What is a Set?
A set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 1, 2, and 3 are distinct objects when considered separately, but when they are considered collectively as the set {1, 2, 3}, they form a single object.
2. Defining a Set
Sets can be defined in two main ways:
- By Extension: Listing all the elements, e.g., A = {a, e, i, o, u}.
- By Comprehension: Describing the properties that its members must satisfy, e.g., A = {x ∈ ℕ | x is even}.
3. Number Sets
Number sets are specific types of sets that include:
- ℕ: Natural numbers
- ℤ: Integers
- ℚ: Rational numbers
- ℝ: Real numbers
- ℂ: Complex numbers
4. Inclusion and Equality
Two sets are equal if they have exactly the same elements. Set A is a subset of set B if all elements of A are also elements of B.
5. Diagrams of Sets
Venn diagrams are used to show all possible logical relations between a finite collection of different sets.
6. Cardinality
The cardinality of a set is a measure of the "number of elements" in the set. For example, if A = {1, 2, 3}, then |A| = 3.
7. Finite or Infinite?
Sets can be finite or infinite. A set is countable if its elements can be counted one by one, even if the process never ends. For example, the set of natural numbers ℕ is countable, while the set of real numbers ℝ is uncountable.
8. Cartesian Product
The Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
9. Venn Diagrams
Venn diagrams are a way of representing sets visually. They show all possible logical relations between a finite collection of different sets.
10. Relations and Functions
A relation from a set A to a set B is a subset of the Cartesian product A × B. A function is a special type of relation where each element of A is related to exactly one element of B.
11. Russell's Paradox
Russell's paradox is a famous problem in set theory that questions whether the set of all sets that do not contain themselves contains itself.
12. Important Laws
- Commutativity: A ∪ B = B ∪ A
- De Morgan's Laws: (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ
Understanding these fundamental concepts of set theory is crucial as they form the foundation of modern mathematics.