Introduction to Sets

Sets

• Sets are well-defined collections of objects.
• Elements within a set are not repeated.
• Sets can be represented using roster or set-builder notation.
• Roster notation lists all elements within curly brackets.
• Set-builder notation describes the elements using a rule.

Types of Sets

• Empty Set (Null Set): A set with no elements (denoted by Ø or {}).
• Finite Set: A set with a countable number of elements.
• Infinite Set: A set with an uncountable number of elements.
• Equal Sets: Sets containing the same elements.
• Subset: Every element of set A is also an element of set B.
• Proper Subset: Set A is a subset of set B, but A is not equal to B.
• Superset: Set B contains all elements of set A.
• Disjoint Sets: Sets with no common elements.

Set Operations

• Union (A∪B): All elements of set A or B (or both).
• Intersection (A∩B): Common elements in sets A and B.
• Difference (A–B): Elements in A but not in B.
• Complement (A'): Elements that are not in A (within a universal set).

Intervals

• Closed Interval [a, b]: Includes a and b.
• Open Interval (a, b): Does not include a or b.
• Half-Open/Half-Closed Interval: Includes or excludes one endpoint.

Venn Diagrams

• Visual representation of sets and their relationships.
• Used to show set operations (union, intersection, difference).

Power Set

• Set of all subsets of a given set.
• Number of elements in a power set with n elements = 2ⁿ.

De Morgan's Laws

• (A∪B)' = A'∩B'
• (A∩B)' = A'∪B'

Cardinality

• Number of elements in a set (denoted as n(A)).

Set Properties

• Commutative property for union and intersection
• Associative property for union and intersection
• Distributive properties relating to intersections and unions.

Practical Applications of Sets

• Practical problems involve applying set theory to real-world contexts e.g., in daily life or in other subjects like probability and statistics.