### Set Theory

**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**:

- Reflexive property: A ⊂ A for any set A.
- 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’ = ∅.