Types of Sets

Understanding the different types of sets is key in the theory of sets. From the empty set to infinite sets, each type plays a unique role in mathematics, helping us categorize and analyze elements in various contexts.

1. Empty Set

   • Denoted by the symbol ∅ or {}.
   • Contains no elements; its cardinality is zero.
   • It is a subset of every set.

2. Singleton Set

   • A set that contains exactly one element.
   • Denoted as {a}, where 'a' is the single element.
   • It is a subset of itself and the empty set is not a singleton.

3. Finite Set

   • A set with a limited number of elements.
   • The cardinality can be counted (e.g., {1, 2, 3} has 3 elements).
   • Examples include {a, b, c} or {1, 2, 3, 4, 5}.

4. Infinite Set

   • A set that has no limit to the number of elements.
   • Cannot be counted completely (e.g., the set of all integers).
   • Examples include the set of natural numbers or real numbers.

5. Universal Set

   • The set that contains all possible elements relevant to a particular discussion.
   • Denoted by the symbol U.
   • Every other set in the context is a subset of the universal set.

6. Subset

   • A set A is a subset of set B if all elements of A are also in B.
   • Denoted as A ⊆ B.
   • The empty set is a subset of every set.

7. Proper Subset

   • A set A is a proper subset of set B if A is a subset of B and A is not equal to B.
   • Denoted as A ⊂ B.
   • Proper subsets must have fewer elements than the set they are compared to.

8. Power Set

   • The set of all possible subsets of a set, including the empty set and the set itself.
   • If a set has n elements, its power set has 2^n elements.
   • Denoted as P(A) for a set A.

9. Disjoint Sets

   • Two sets are disjoint if they have no elements in common.
   • Denoted as A ∩ B = ∅.
   • Examples include {1, 2} and {3, 4}.

10. Equal Sets

    • Two sets are equal if they contain exactly the same elements.
    • Denoted as A = B.
    • Order of elements does not matter in set equality.

11. Equivalent Sets

    • Two sets are equivalent if they have the same cardinality, regardless of the actual elements.
    • Denoted as |A| = |B|.
    • Example: {1, 2} and {a, b} are equivalent sets.

12. Countable Sets

    • A set is countable if its elements can be put into a one-to-one correspondence with the natural numbers.
    • Includes both finite sets and infinite sets like the set of integers.
    • Examples include {1, 2, 3} and {0, 1, 2, ...}.

13. Uncountable Sets

    • A set that cannot be put into a one-to-one correspondence with the natural numbers.
    • Examples include the set of real numbers between any two integers.
    • Uncountable sets have a greater cardinality than countable sets.

14. Open Sets

    • A set is open if it contains none of its boundary points.
    • Commonly used in topology; for example, the interval (a, b) is open.
    • Open sets can be infinite and are essential in defining continuity.

15. Closed Sets

    • A set is closed if it contains all its boundary points.
    • For example, the interval [a, b] is closed.
    • Closed sets are important in analysis and topology, particularly in limit points.