Basic Set Notations
In mathematics, sets are collections of distinct objects, called elements or members. Sets are typically represented using braces { }, and the elements of a set are listed inside the braces, separated by commas. Here are some commonly used notations, symbols, and their meanings related to sets:
1. Set Notation:
List notation: A set can be defined by explicitly listing its elements. For example, A = {1, 2, 3} represents a set named A that contains the elements 1, 2, and 3.
Set-builder notation: A set can be defined by specifying a rule or condition that its elements must satisfy. For example, B = {x | x is an even number} represents a set named B that contains all even numbers.
2. Set Symbols:
∈ (element of): This symbol is used to indicate that an element belongs to a set. For example, if x is an element of set A, it is denoted as x ∈ A.
∉ (not an element of): This symbol is used to indicate that an element does not belong to a set. For example, if x is not an element of set A, it is denoted as x ∉ A.
⊂ (subset): This symbol is used to represent that one set is a subset of another set. For example, if every element of set A is also an element of set B, it is denoted as A ⊂ B.
⊆ (subset or equal to): This symbol is used to represent that one set is a subset of or equal to another set. For example, if every element of set A is also an element of set B (possibly including the case where A and B are identical), it is denoted as A ⊆ B.
∅ (empty set): This symbol represents the empty set, which is a set with no elements.
3. Set Operations:
Union: The union of two sets A and B, denoted by A ∪ B, is a set that contains all the elements that are in A or in B (or in both).
Intersection: The intersection of two sets A and B, denoted by A ∩ B, is a set that contains all the elements that are in both A and B.
Complement: The complement of a set A, denoted by A', is a set that contains all the elements that are not in A, usually with respect to a given universal set.
Difference: The difference of two sets A and B, denoted by A - B or A \ B, is a set that contains all the elements that are in A but not in B.
These are some of the basic notations, symbols, and meanings used in set theory. There are other advanced concepts and symbols used in set theory as well, but the ones mentioned above are fundamental.