Sets and Their Definitions

What is a Set?

• A set is a collection of well-defined, distinct objects.
• Objects in a set are called elements.

Well-defined sets

• It is possible to determine if an object belongs to a well-defined set.
• Examples:
  • A set of fruits.
  • A set of counting numbers.
  • A set of even numbers.

Not Well-defined Sets

• It is not possible to determine if an object belongs to a set that is not well-defined.
• Examples:
  • The set of famous dancers.
  • The set of honest people.
  • The set of good writers.

Describing Sets

• Verbal Description Method: The set is described in words using a verbal statement.
• Roaster or Listing Method: The elements of the set are listed in a row separated by commas and enclosed with braces.
• Set Builder Notation: A mathematical notation for describing a set where you state the properties that its members must satisfy.

Types of Sets

• Finite Sets: A set with a limited number of elements that can be counted.
• Infinite Sets: A set with an unlimited number of elements that may or may not be counted.
• Empty or Null Sets: A set with no elements. The symbol is {} or { }.
• Universal Set: A set containing all relevant elements for a particular context. It is denoted by the symbol 'U'.

Cardinality of a Set

• The Cardinal number of a set 'A', denoted by N(A), is the number of elements in the set.
• It refers to the number of elements present in the set.

Sets and their elements

• A set is a collection of distinct objects called elements.
• Sets can be described in multiple ways:
  • Verbal description method: Describes the elements using words.
  • Listing method: Lists all the elements within curly braces.
  • Set builder notation: Uses a statement to describe the elements.

Cardinality of a set

• The cardinality of a set is the number of its elements.

Subsets

• A set A is a subset of another set B if every element in A is also an element in B.
• This is represented using the symbol: ⊆ (read as "is a subset of").
• Every set is a subset of itself.
• The empty set (∅) is a subset of every set.

Union of sets

• The union of sets A and B (written as A ∪ B) includes all elements from both A and B.
• It is represented using the symbol: ∪ (read as "union").

Intersection of sets

• The intersection of sets A and B (written as A ∩ B) includes only the elements that are present in both A and B.
• It is represented using the symbol: ∩ (read as "intersection").