Sets in Mathematics

Sets

• Sets are well-defined collections of objects.
• Elements within a set are not repeated.
• Sets are represented using capital letters and curly brackets.
• Example: A = {1, 2, 3, 4}

Types of Sets

• Empty Set (Null Set): A set with no elements. Represented as Ø or {}.
• Finite Set: A set with a countable number of elements. Example: {1, 2, 3}.
• Infinite Set: A set with an uncountable number of elements. Example: The set of natural numbers.
• Equal Set: Sets with precisely the same elements. Example: A = {1, 2, 3} and B = {1, 2, 3}
• Subset: A set A is a subset of set B, written as A ⊆ B, if every element of A is also an element of B.
  • Example: {1, 2} ⊆ {1, 2, 3}
• Superset: Set B is a superset of set A, written as B ⊇ A if every element of A is also an element of B.
• Proper Subset: Set A is a proper subset of set B, written as A ⊂ B, if A is a subset of B, but A is not equal to B.
• Roster Form: Lists all elements within curly brackets.
• Set-Builder Notation: Defines a set using a rule or property that determines set membership. -Example: {x | x is a natural number less than 5}.

Set Operations

• Union (A∪B): Includes all elements from both sets A and B. A∪B = {x | x ∈ A or x ∈ B}
• Intersection (A∩B): Contains elements common to both sets A and B. A∩B = {x | x ∈ A and x ∈ B}
• Difference (A-B): Contains elements in set A that are not in set B. A-B = {x | x ∈ A and x ∉ B}
• Complement (A'): Contains all elements in the universal set (U) that are not in set A. A'= U - A

Set Properties

• Commutative Law: A∪B = B∪A and A∩B = B∩A.
• Associative Law: (A∪B)∪C = A∪(B∪C) and (A∩B)∩C = A∩(B∩C).
• Distributive Law: A∩(B∪C) = (A∩B)∪(A∩C) and A∪(B∩C) = (A∪B)∩(A∪C).
• Idempotent Law: A∪A = A and A∩A = A.
• Identity Law: A∪$ = A and A∩U = A.
• Complement Laws: A∪A’=U and A∩A’ = $.
• De Morgan's Laws: (A∪B)’ = A’∩B’ and (A∩B)’ = A’∪B’.

Intervals

• Intervals are subsets of real numbers
• Example of closed interval: [a, b]={x: a ≤ x ≤ b}
• Example of open interval: (a, b)={x: a < x < b}

Cardinality

• n(A): the number of elements in set A
• n(A∪B) = n(A) + n(B) - n(A∩B)

Venn Diagrams

• Visual representations of sets and their relationships, using overlapping circles.

Additional Information

• Set of natural numbers (N): {1, 2, 3, ...}
• Set of whole numbers (W): {0, 1, 2, 3, ...}
• Set of integers (Z): {..., -3, -2, -1, 0, 1, 2, 3, ...}
• Set of rational numbers (Q): Numbers that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
• Set of real numbers (R): Includes all rational and irrational numbers.
• Set of complex numbers (C): Includes real and imaginary numbers.