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Theory. A set of ideas that explain something. In Mathematics a theory is the set of theorems and principles that make up a branch of mathematics, such as Number Theory, Set Theory, etc. In Science a theory is well understood, has been properly examined, and can be tested again and again and always works, such as the Theory of Gravity.
Reply. ex0du5 •. "Theory" in math is a precise formal term to indicate a set of rules (axioms) and their logical calculus. It is the syntactical definition of a particular symbolic game. In this meaning, we have things like ZFC and NF being very typical examples of theories.
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- Fundamental set concepts
set theory, branch of mathematics that deals with the properties of well-defined collections of objects, which may or may not be of a mathematical nature, such as numbers or functions. The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the definition of complex and sophisticated mathematical concepts.
Between the years 1874 and 1897, the German mathematician and logician Georg Cantor created a theory of abstract sets of entities and made it into a mathematical discipline. This theory grew out of his investigations of some concrete problems regarding certain types of infinite sets of real numbers. A set, wrote Cantor, is a collection of definite, distinguishable objects of perception or thought conceived as a whole. The objects are called elements or members of the set.
The theory had the revolutionary aspect of treating infinite sets as mathematical objects that are on an equal footing with those that can be constructed in a finite number of steps. Since antiquity, a majority of mathematicians had carefully avoided the introduction into their arguments of the actual infinite (i.e., of sets containing an infinity of objects conceived as existing simultaneously, at least in thought). Since this attitude persisted until almost the end of the 19th century, Cantor’s work was the subject of much criticism to the effect that it dealt with fictions—indeed, that it encroached on the domain of philosophers and violated the principles of religion. Once applications to analysis began to be found, however, attitudes began to change, and by the 1890s Cantor’s ideas and results were gaining acceptance. By 1900, set theory was recognized as a distinct branch of mathematics.
At just that time, however, several contradictions in so-called naive set theory were discovered. In order to eliminate such problems, an axiomatic basis was developed for the theory of sets analogous to that developed for elementary geometry. The degree of success that has been achieved in this development, as well as the present stature of set theory, has been well expressed in the Nicolas Bourbaki Éléments de mathématique (begun 1939; “Elements of Mathematics”): “Nowadays it is known to be possible, logically speaking, to derive practically the whole of known mathematics from a single source, The Theory of Sets.”
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Numbers and Mathematics
In naive set theory, a set is a collection of objects (called members or elements) that is regarded as being a single object. To indicate that an object x is a member of a set A one writes x ∊ A, while x ∉ A indicates that x is not a member of A. A set may be defined by a membership rule (formula) or by listing its members within braces. For example, the set given by the rule “prime numbers less than 10” can also be given by {2, 3, 5, 7}. In principle, any finite set can be defined by an explicit list of its members, but specifying infinite sets requires a rule or pattern to indicate membership; for example, the ellipsis in {0, 1, 2, 3, 4, 5, 6, 7, …} indicates that the list of natural numbers ℕ goes on forever. The empty (or void, or null) set, symbolized by {} or Ø, contains no elements at all. Nonetheless, it has the status of being a set.
A set A is called a subset of a set B (symbolized by A ⊆ B) if all the members of A are also members of B. For example, any set is a subset of itself, and Ø is a subset of any set. If both A ⊆ B and B ⊆ A, then A and B have exactly the same members. Part of the set concept is that in this case A = B; that is, A and B are the same set.
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Two sets A and B are equal if and only if A ⊂ B and B ⊂ A. If A ⊂ B and A does not equal B, we say that A is a proper subset of B, and write A ⊊ B. The set θ = {x: x ≠ x} is called the empty set. This set clearly has no elements. Using Theorem 1.1.1, it is easy to show that all sets with no elements are equal.
Set Theory is a branch of mathematical logic where we learn sets and their properties. A set is a collection of objects or groups of objects. These objects are often called elements or members of a set. For example, a group of players in a cricket team is a set. Since the number of players in a cricket team could be only 11 at a time, thus we ...
- Set theory defines the collection of objects, where the order of objects does not matter. It relates with the collection of group of members or ele...
- The different types of sets are finite and infinite sets, subset, power set, empty set or null set, equal and equivalent sets, proper and improper...
- If we have two sets, P and Q, then, n(P U Q) = n(P) + (n(Q) – n (P ⋂ Q), where n denotes the number of elements. For three sets, P, Q and R, the...
- A singleton set has only one element. Example are {1}, {a}, {x:x is an integer that has only one factor}, etc.
- Sets can be represented in two different ways: Roster form and Set-builder form
Theoretical mathematics is the study of abstract mathematical structures which form the basic framework for the rest of the mathematical sciences. In large part, theoretical mathematics is inspired by intellectual curiosity. Theoretical mathematics provides the tools for scientific discoveries in the future, often in unexpected ways.
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Many mathematical concepts would be difficult to define precisely (and concisely) without the use of set theory. As such, it is important to be familiar with the various symbols and notations used in set theory in order to both understand and communicate mathematical concepts effectively. The table below includes some of the most common symbols.
