Numeracy

The Basics of Simple Set Theory




Set theory is the study of collections of objects, known as sets. It forms a fundamental building block for advanced mathematics. Here's a breakdown of the basics:

Key Definitions

  1. Set: A collection of objects with something in common.

  2. Example: Prime numbers or types of birds.

  3. Elements/Members: Objects within a set.

  4. Example: In the set of prime numbers {2, 3, 5}, the elements are 2, 3, and 5.

  5. Symbols:

  6. ?: "Is an element of." (e.g., ( 2 \in A ))

  7. ?: "Is not an element of." (e.g., ( 10 A ))

  8. Notation:

  9. Sets are written with curly brackets, e.g., ( A = {1, 2, 3} ).

  10. Use ellipses (e.g., ( {1, 2, \dots, 10} )) for long sequences but ensure clarity.

  11. Cardinality: The number of elements in a set.

  12. Example: The cardinality of ( A = {1, 2, 3} ) is 3.

  13. Subset (?): A set whose elements are all contained within another set.

  14. Example: ( A = {1, 2}, B = {1, 2, 3}, A B ).

  15. Empty Set (Ø): A set with no elements.

  16. Always a subset of any set.

  17. Universal Set (U): The "everything" set for a specific context.

  18. Example: For numbers between 1 and 10, ( U = {1, 2, \dots, 10} ).

Set Operations

  1. Union (?): Combines all elements from two sets.
  2. Example: ( A = {1, 2}, B = {2, 3}, A B = {1, 2, 3} ).

  3. Rules:

    • Duplicates are included only once.
    • ( A A = A ), ( A = A ).

  4. Intersection (?): Finds elements common to both sets.

  5. Example: ( A = {1, 2}, B = {2, 3}, A B = {2} ).

  6. If no elements are shared, the intersection is the empty set (( {} )).

  7. Relative Complement (?): Removes elements in one set that are also in another.

  8. Example:

    • ( A = {1, 2}, B = {2, 3}, A - B = {1} ).
    • ( B - A = {3} ).

  9. Complement (A?): Includes all elements in the universal set not in the specified set.

  10. Example: ( U = {1, 2, 3, 4}, A = {1, 2}, A? = {3, 4} ).

Quick Tips and Warnings

  • Unambiguous Ellipses: Clearly define the pattern in your set when using ellipses.
  • Identical Sets: Sets are identical if they have the same elements, regardless of order or duplication.
  • Example: ( {Monday, Tuesday} = {Tuesday, Monday} ).
  • Use in Higher Mathematics: Though sets may seem basic, they underpin many advanced mathematical concepts.

? Understanding the basics of set theory ensures a strong foundation for tackling more complex mathematical ideas later.

❤ If you liked this, consider supporting us by checking out Tiny Skills - 250+ Top Work & Personal Skills Made Easy