Given a binary relation R on sets A and B, we say that the R is a function if each element of A is paired with at most one element of B. In this case we call A the domain set, and we call B the range set. If f is such a function and the ordered pair (x,y) is an element of the function, we typically write f(x) = y.
More generally, if R is an n-place relation, then R is a function of n-1 variables if each (n-1)-tuple is matched with at most one element of the range set (i.e., the nth set of the Cartesian product on which the relation is defined). If in addition there is at most one element of the domain set matched with any given element of the range set, then the function is called “one-to-one” or injective. If the function maps at least one element of the domain set to every element of its range set, then it is called “onto” or surjective. A function which is both injective and surjective is called bijective. Functions with range in the real numbers or complex numbers are called real-valued or complex-valued respectively.
See predicate calculus.
Fundamental Theorem of Algebra
The field of complex numbers is
algebraically closed, that is, any polynomial with complex coefficients has a complex root.
Fundamental Theorem of Arithmetic
Every natural number is either prime or may be decomposed uniquely into prime factors.
Fundamental Theorem of Calculus
Intitively, that the integral and derivative are inverse operatorations on functions. The fundamental theorem is commonly expressed in either one of two formal guises:
Cf. Riemann integral.
Italian mathematician, physicist, and astronomer who is generally associated with the birth of modern science. Such discoveries as that bodies of different size and weight fall at the same speed (i.e., that gravitational acceleration is independent of mass), overthrew two millenia of Aristotelian doctrines. They also greatly annoyed certain factions within the Church, which was only just coming to terms with the birth of science, and ultimately brought about a showdown with ecclesiastical authorities that effectively silenced Galileo at the end of his life. Galileo’s importance in the history of mathematics stems less from his purely mathematical work than from his promulgation of an outlook, especially in the sciences, that intellectual inquiry must not be fettered by prejudice if it is to bear fruit.
Abbreviation for greatest common divisor.
See: generalized continuum hypothesis.
If a is a complex number that is also algebraic, and if b is an irrational number, then a b is a transcendental number.
generalized continuum hypothesis
The claim that there is no set of intermediate cardinality between any given set and its power set. The GCH is independent of (i.e., cannot be proved true or false by) the known axioms of set theory.
Cf. continuum hypothesis.
Topology: Given a topological surface S, the genus of S is the ordered pair (h, c) in which h is the number of handles on the surface and c is the number of cross-caps.
Graph Theory: The genus of a graph is the minimum genus of the topological surface on which the graph can be drawn without crossovers. Intuitively, the minumum number of handles one must attach to a sphere to draw the graph on it.
A curve on a space is a geodesic if, given any two points on the curve, the segment of the curve joining the points is the shortest such curve. In Euclidean space, geodesics are straight lines. On a sphere, the geodesics are the great circles of the sphere.
Graph Theory: A shortest path between two vertices is sometimes called a geodesic.
An infinite series such that the ratio of consecutive terms is constant. Such a series may be denoted
where r is the common ratio. A geometric series converges if and only if the common ratio is less than 1, in which case the sum of the series is given by
Cf. arithmetic series, harmonic series.
Abbreviation for greatest lower bound.
The foremost logician of the 20th century. He is famous for many deep results in the foundations of mathematics, including the Godel Completeness Theorem, the Godel Incompleteness Theorem, and that the generalized continuum hypothesis and the axiom of choice are consistent with the other axioms of set theory. He moved from Austria to the Institute for Advanced Study at Princeton, New Jersey, in 1940. He was a lifelong friend of Albert Einstein, and in the 1950’s formulated versions of Einstein’s relativity theories that permitted the logical possibility of time travel.
General: A picture or diagram representing one or more relationships among certain objects or quantities.
Functions: The collection of ordered tuples (x,y), consisting of domain elements x (possibly also tuples) and corresponding range elements y. Although informally we often mean the picture when we speak of the graph of a function, the graph should always be formally thought of as a set of tuples.
Graph Theory: A collection of nodes (also called vertices), together with a set of edges joining the nodes. Formally, a graph G is a set of vertices V, a set of edges E, and a symmetric relation between them specifying incidence. Every edge must be incident on exactly two (not necessarily distinct) vertices, called its ends. Two vertices incident on the same edge are called adjacent. Graphs are often represented pictorially, with the vertices being points (or small circles or squares or some such), and the edges being curves (often line segments), with the endpoints of each such curve being the ends of that edge in the graph.
Cf. Konigsberg Bridge Problem.
On a sphere, a great circle is a circle containing the endpoints of a diameter of the sphere, that is, a circle of maximal size. A geodesic on a sphere always lies on a great circle.
greatest common divisor
The greatest common divisor of two natural numbers a and b is the largest natural number c such that c divides a and c divides b.
greatest lower bound
A lower bound which is greater than or equal to every other lower bound.
Cf. Least upper bound.
Letters of the Greek alphabet are commonly used in mathematics. See the article for a full description.
A set together with a binary operation defined on its elements, satisfying
If the group operation is commutative, the group is called Abelian. If the condition that every element have an inverse is dropped, the set is called a monoid. If in addition the requirement that an identity element exist is dropped, the set is called a semigroup. Finally, if the condition that the group operation be associative is dropped, the set is called a groupoid.
- The group operation is associative,
- There is an identity element, i.e., an element e of the group so that for any other element a of the group we have ea = ae = a, and
- for every element a of the group there is an element a´, called the inverse of a, so that aa´ = a´a = e.
Cf. field, module, ring