15-09-2017, 03:53 PM
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected in a set, set theory is most often applied to objects that are relevant to mathematics. The language of set theory can be used in the definitions of almost all mathematical objects.
The modern study of set theory was pioneered by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in the theory of naive sets, such as Russell's paradox, numerous systems of axioms were proposed at the beginning of the century XX, of which the axioms of Zermelo-Fraenkel, with or without the axiom of choice, are the best known.
Set theory is commonly used as a foundational system for mathematics, particularly in the form of Zermelo-Fraenkel's set theory with the axiom of choice. Beyond its fundamental role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research in set theory includes a diverse collection of topics, ranging from the structure of the line of real numbers to the study of the consistency of the great cardinals.
Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member (or element) of A, the notation or ∈ A is used. Since sets are objects, the membership relation also can relate sets.
A binary derived relationship between two sets is the subset relationship, also called set inclusion. If all members of set A are also members of set B, then A is a subset of B, denoted A ⊆ B. For example, {1, 2} is a subset of {1, 2, 3}, and so is {2} but {1, 4} is not. As is implied by this definition, a set is a subset of itself. For cases where this possibility is not adequate or makes sense to reject, the term appropriate subset is defined. A is called a proper subset of B if and only if A is a subset of B, but A is not equal to B. Note also that 1 and 2 and 3 are members (elements) of the set {1, 2, 3} but are not subsets, and subsets, as {1}, in turn, are not members of the set {1, 2, 3}.
Just as arithmetic presents binary operations on numbers, set theory presents binary operations in sets. The:
• Union of sets A and B, called A ∪ B, is the set of all objects that are members of A, or B, or both. The union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4}.
• The intersection of sets A and B, denoted A ∩ B, is the set of all objects that are members of A and B. The intersection of {1, 2, 3} and {2, 3, 4} is set {2, 3}.
• The established difference of U and A, denoted U \ A, is the set of all members of U that are not members of A. The difference established {1, 2, 3} {2, 3, 4} is { 1, whereas, conversely, the established difference {2, 3, 4} {1, 2, 3} is {4}. When A is a subset of U, the set difference U \ A is also called the complement of A in U. In this case, if the choice of U is clear from the context, sometimes the notation Ac is used instead of U \ A, particularly if U is a universal set as in the study of Venn diagrams.
• The symmetric difference of sets A and B, called A △ B or A ⊖ B, is the set of all objects that are a member of exactly one of A and B (elements that are in one of the sets, but not in both) . For example, for sets {1, 2, 3} and {2, 3, 4}, the symmetric difference is {1, 4}. It is the established difference of the junction and intersection, (A ∪ B) \ (A ∩ B) or (A \ B) ∪ (B \ A).
• The Cartesian product of A and B, denoted A × B, is the set whose members are all possible pairs ordered (a, b) where a is a member of A and b is a member of B. The Cartesian product of {1, 2 and {red, white} is {(1, red), (1, white), (2, red), (2, white)}.
• The power set of a set A is the set whose members are all possible subsets of A. For example, the power set of {1, 2} is {{}, {1}, {2}, {1 , 2}} .
Some basic sets of central importance are the empty set (the unique set that does not contain elements, occasionally called the null set although this name is ambiguous), the set of natural numbers and the set of real numbers.
The modern study of set theory was pioneered by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in the theory of naive sets, such as Russell's paradox, numerous systems of axioms were proposed at the beginning of the century XX, of which the axioms of Zermelo-Fraenkel, with or without the axiom of choice, are the best known.
Set theory is commonly used as a foundational system for mathematics, particularly in the form of Zermelo-Fraenkel's set theory with the axiom of choice. Beyond its fundamental role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research in set theory includes a diverse collection of topics, ranging from the structure of the line of real numbers to the study of the consistency of the great cardinals.
Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member (or element) of A, the notation or ∈ A is used. Since sets are objects, the membership relation also can relate sets.
A binary derived relationship between two sets is the subset relationship, also called set inclusion. If all members of set A are also members of set B, then A is a subset of B, denoted A ⊆ B. For example, {1, 2} is a subset of {1, 2, 3}, and so is {2} but {1, 4} is not. As is implied by this definition, a set is a subset of itself. For cases where this possibility is not adequate or makes sense to reject, the term appropriate subset is defined. A is called a proper subset of B if and only if A is a subset of B, but A is not equal to B. Note also that 1 and 2 and 3 are members (elements) of the set {1, 2, 3} but are not subsets, and subsets, as {1}, in turn, are not members of the set {1, 2, 3}.
Just as arithmetic presents binary operations on numbers, set theory presents binary operations in sets. The:
• Union of sets A and B, called A ∪ B, is the set of all objects that are members of A, or B, or both. The union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4}.
• The intersection of sets A and B, denoted A ∩ B, is the set of all objects that are members of A and B. The intersection of {1, 2, 3} and {2, 3, 4} is set {2, 3}.
• The established difference of U and A, denoted U \ A, is the set of all members of U that are not members of A. The difference established {1, 2, 3} {2, 3, 4} is { 1, whereas, conversely, the established difference {2, 3, 4} {1, 2, 3} is {4}. When A is a subset of U, the set difference U \ A is also called the complement of A in U. In this case, if the choice of U is clear from the context, sometimes the notation Ac is used instead of U \ A, particularly if U is a universal set as in the study of Venn diagrams.
• The symmetric difference of sets A and B, called A △ B or A ⊖ B, is the set of all objects that are a member of exactly one of A and B (elements that are in one of the sets, but not in both) . For example, for sets {1, 2, 3} and {2, 3, 4}, the symmetric difference is {1, 4}. It is the established difference of the junction and intersection, (A ∪ B) \ (A ∩ B) or (A \ B) ∪ (B \ A).
• The Cartesian product of A and B, denoted A × B, is the set whose members are all possible pairs ordered (a, b) where a is a member of A and b is a member of B. The Cartesian product of {1, 2 and {red, white} is {(1, red), (1, white), (2, red), (2, white)}.
• The power set of a set A is the set whose members are all possible subsets of A. For example, the power set of {1, 2} is {{}, {1}, {2}, {1 , 2}} .
Some basic sets of central importance are the empty set (the unique set that does not contain elements, occasionally called the null set although this name is ambiguous), the set of natural numbers and the set of real numbers.