Set theory

Here’s an explanation of set theory, presented in a notes format with emojis for easy understanding:

Set Theory 📚

• What is a Set?

  • At its most basic level, a set is a collection of items.
  • These items are called elements or members.
  • Sets can be finite (e.g., days of the week 🗓️, months in a year) or infinite (e.g., the set of integers 🔢).

• Key Characteristics of Sets ✨

  • Order is not important 🔄: Listing elements in a different sequence does not change the set (e.g., {Kohli, Dhoni} is the same as {Dhoni, Kohli}).
  • Duplicates do not matter 🚫: Including the same element multiple times does not change the set (e.g., {1, 2, 2, 3} is the same as {1, 2, 3}).

• Cardinality (Size of a Set) 📏

  • The cardinality of a set is the number of items it contains.
  • For finite sets, you can simply count the elements.

• Important Sets of Numbers 🔢

  • In mathematics, we commonly deal with various infinite sets of numbers:
    • Natural Numbers (N): {0, 1, 2, 3, …} (often includes 0 in this context).
    • Integers (Z): {…, -2, -1, 0, 1, 2, …} (natural numbers extended with negative numbers).
    • Rational Numbers (Q): Numbers that can be written as a fraction p/q where p and q are integers (e.g., 1/2, 7/1).
    • Real Numbers (R): All rational numbers plus irrational numbers (like √2, π).

• Relationships Between Sets 🤝

  • Element of (∈): Used to show that an item belongs to a set (e.g., 5 ∈ Z means 5 is an integer).
  • Not an Element of (∉): Indicates an item does not belong to a set (e.g., √2 ∉ Q means √2 is not a rational number).
  • Subset (⊆): A set X is a subset of set Y if every element in X is also an element in Y (e.g., N ⊆ Z, meaning all natural numbers are also integers).
  • Proper Subset (⊂): X is a proper subset of Y if X is a subset of Y, but X is not equal to Y (meaning Y contains at least one element not in X).
  • Equality of Sets (=): Two sets X and Y are equal if and only if they contain exactly the same elements. This can be proven by showing X ⊆ Y and Y ⊆ X.
  • Venn Diagrams 🖼️: Useful pictorial representations of sets and their relationships, showing how sets are included in others or how they overlap.

• Special Sets 🎁

  • Empty Set (∅ or {}): The set that contains no elements.
    • It is a subset of every set.
  • Power Set (𝒫(X)): The set of all possible subsets of a given set X.
    • For a finite set with n elements, its power set will have 2^n subsets.

• Set Comprehension (Building Subsets) 🛠️

  • This is a notation used to define subsets, especially for infinite sets where listing all elements is impossible.
  • It generally involves three parts:
    • Generator: Specifies the existing set from which elements are taken (e.g., x ∈ Z means ‘for every x in the set of integers’).
    • Filter: A condition that elements must satisfy to be included in the new set (e.g., x mod 2 = 0 means ‘x must be even’).
    • Transformer: An operation performed on the filtered elements to produce the new set’s elements (e.g., x² means ‘square the number’). Sometimes the transformer is just to keep the element as it is.
  • Example: {x² | x ∈ Z, x mod 2 = 0} means “the set of squares of all even integers”.

• Operations on Sets ➕➖✖️➗

  • Union (X ∪ Y): Combines all elements from two sets X and Y into a single set, removing any duplicates.
  • Intersection (X ∩ Y): Contains only the elements common to both sets X and Y.
  • Set Difference (X \ Y or X - Y): Contains elements that are in set X but not in set Y. The order of sets matters for difference.
  • Complement (Xᶜ or X’): Consists of all elements not in set X.
    • It requires a defined “universe” or overall set from which elements are drawn. Without a universe, “complement” is ambiguous (e.g., “numbers that are not prime” needs to specify if it means non-prime natural numbers, integers, or reals).

• Foundation of Mathematics 🏛️

  • Set theory was developed to be a foundation for all of mathematics.
  • However, certain “collections” like “the set of all sets” lead to paradoxes (e.g., Russell’s Paradox), indicating that not every collection can logically be considered a set. This highlights the need for careful definitions in set theory.
  • Sets are fundamental for understanding and defining various mathematical concepts, including different types of numbers and relations.