What are the definitions of sets theory
(1) subsets
(2) De Morgan's Laws
(3) Sets operation
(1) Subsets: In set theory, a subset refers to a set that consists of elements from another set. Formally, if every element of set A is also an element of set B, then A is considered to be a subset of B, denoted as A ⊆ B. It can also be represented as A is a subset of B if for all x, if x is in A, then x is in B.
(2) De Morgan's Laws: De Morgan's Laws are a pair of fundamental laws in set theory that describe the relationships between the complement of sets and set operations. There are two laws: the first law states that the complement of the union of two sets is equal to the intersection of their complements, while the second law states that the complement of the intersection of two sets is equal to the union of their complements.
First Law (De Morgan's Law): For any two sets A and B, the complement of the union of A and B is equal to the intersection of their complements. It is represented as:
(A ∪ B)' = A' ∩ B'
Second Law (De Morgan's Law): For any two sets A and B, the complement of the intersection of A and B is equal to the union of their complements. It is represented as:
(A ∩ B)' = A' ∪ B'
(3) Sets Operation: Sets operations are mathematical operations that can be performed on sets to create new sets. The most common sets operations include:
- Union: The union of two sets A and B is a new set that contains all the elements that are in either A or B (or in both). It is denoted by A ∪ B.
- Intersection: The intersection of two sets A and B is a new set that contains all the elements that are common to both A and B. It is denoted by A ∩ B.
- Complement: The complement of a set A, denoted by A', is a new set that contains all the elements that are not in A but are in the universal set of reference. For example, if the universal set is U, then A' contains all the elements in U that are not in A.
- Difference: The difference between two sets A and B, denoted by A - B, is a new set that contains all the elements that are in A but are not in B.