Sets

Definition: A set is a collection of distinct objects. Representation: Sets are represented using curly braces {}. Elements: The elements of a set are separated by commas.

Subsets

Definition: A subset of a set is a set that contains all of the elements of the original set. Representation: The subset of a set A is represented as A ⊂ B. Examples: A ⊂ {1, 2, 3} means that set A is a subset of the set {1, 2, 3}. Properties:

1. Reflexive property: A ⊂ A for any set A.
2. Transitive property: If A ⊂ B and B ⊂ C, then A ⊂ C.

Union

Definition: The union of two sets is a set that contains all of the elements of both sets. Representation: The union of two sets A and B is represented as A ∪ B. Examples: A ∪ {1, 2, 3} = {1, 2, 3, 4, 5} means that the union of set A and the set {1, 2, 3} is the set {1, 2, 3, 4, 5}.

Intersection

Definition: The intersection of two sets is a set that contains only the elements that are common to both sets. Representation: The intersection of two sets A and B is represented as A ∩ B. Examples: A ∩ {1, 2, 3} = {2} means that the intersection of set A and the set {1, 2, 3} is the set {2}.

Complement

Definition: The complement of a set A with respect to a universal set U is a set that contains all the elements of U that are not in A. Representation: The complement of a set A is represented as A’. Examples: If the universal set U = {1, 2, 3, 4, 5} and A = {1, 2, 3}, then A’ = {4, 5} means that the complement of set A with respect to the universal set U is the set {4, 5}.

Power Set

Definition: The power set of a set A is the set of all subsets of A. Representation: The power set of a set A is represented as P(A). Examples: If A = {1, 2, 3}, then P(A) = {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} means that the power set of the set A is the set of all subsets of A, including the empty set.

Cardinality

Definition: The cardinality of a set is the number of elements in the set. Representation: The cardinality of a set A is represented as |A|. Examples: If A = {1, 2, 3}, then |A| = 3 means that the cardinality of the set A is 3.

Properties of Sets

Commutative property: A ∪ B = B ∪ A and A ∩ B = B ∩ A. Associative property: (A ∪ B) ∪ C = A ∪ (B ∪ C) and (A ∩ B) ∩ C = A ∩ (B ∩ C). Distributive property: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). Identity property: A ∪ ∅ = A and A ∩ U = A. Complement property: A ∪ A’ = U and A ∩ A’ = ∅.