Sets and Their Properties

Sets

• A set is a well-defined collection of objects.
• Well-defined means each object in the collection is clearly identifiable.
• Elements of a set are distinct

Representation of Sets

• Roster Method (Tabular Form): Listing all elements within curly brackets. Examples:
  • Vowels: {a, e, i, o, u}
  • First five natural numbers: {1, 2, 3, 4, 5}
• Set-builder Form: Defining the set by a property shared by all elements. Example:
  • Set of even natural numbers: {x: x is an even natural number}

Classification of Sets

• Finite Set: Contains a countable number of elements. Example: The set of letters in the English alphabet.
• Infinite Set: Contains an uncountable number of elements. Example: The set of natural numbers.
• Empty (Null) Set: A set with no elements, denoted by ∅ or {}.
• Singleton Set: A set with only one element. Example: {2}

Subsets

• A set A is a subset of set B (A ⊂ B) if every element of A is also an element of B.
• The null set is a subset of every set.

Union of Sets

• The union of two sets A and B (A ∪ B) is a set containing all elements from either set A, or set B, or both.

Intersection of Sets

• The intersection of two sets A and B (A ∩ B) is a set containing only the elements common to both set A and set B.
• Disjoint sets have no elements in common (A ∩ B = ∅).

Complement of a Set

• The complement of a set A (A') is the set of all elements in the universal set that are not in set A.
• A' = U – A. U represents the universal set containing all elements under consideration.

Venn Diagrams

• Visual representations of sets using overlapping circles.
• Represent relationships between sets.

Explore the fundamentals of sets, including their definitions, representations, and classifications. This quiz covers concepts like finite and infinite sets, subsets, and examples to enhance your understanding of set theory.