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Dodgson, Charles Lutwidge – extended real numbers
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Lewis Carroll
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Dodgson, Charles Lutwidge
(born 1832)
Oxford mathematician most famous for the books Alice in Wonderland and Through the Looking Glass, which he wrote under the pseudonym Lewis Carroll. However, he also wrote several mathematics textbooks, and delighted in inventing bizarre and humorous syllogisms for exercises in Aristotelian logic. These syllogisms commonly appear in introductory texts on logic to this day. He also formulated what is now called Carroll's Paradox, which shows the need for formal rules of inference in any system of logic.
Related article: Carroll's Paradox
domain
General: A universe of discourse, that is, the class of objects under consideration. Functions and relations: The domain of a function (relation) is the set of elements which the function (relation) maps to its range set.
e
See Euler number.
edge
General: A line formed by the intersection of two planes. In a 3-dimensional figure (such as a polyhedron), the line or curve where two faces or surfaces meet.
Graph Theory: One of two kinds of entities in a graph. Restricted to being incident on exactly two vertices.
edge set
The set of edges of some graph. For a graph G, the edge set of G is denoted by E(G), or, if there is no ambiguity as to the graph in question, simply by E.
ellipse
The locus of points in the plane, the sum of whose distances from two fixed points, called the foci, remains constant.
Like the hyperbola and parabola, the ellipse is a conic section.
Related article: Conics
empty set
The unique set having no elements, generally denoted by a circle with a forward slash through it, or by an empty pair of braces.
Epimenides Paradox
See Liar Paradox.
epimorphism
A morphism f from X to Y is called an epimorphism when it is surjective, that is, when to each element y of Y there corresponds an x in X such that f(x) = y.
Cf. monomorphism.
equilateral triangle
A triangle with equal sides and equal angles.
equipotent
Two sets are equipotent if there exists a function between them that is bijective, that is, which is “one-to-one” and “onto.”
Cf. cardinal.
equivalence relation
An equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive. An equivalence relation on X gives rise to (and is determined by) a partition on X.
Cf. congruence relation.
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self portrait
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Escher, Maurits Cornelis
(born 1898)
Dutch graphic artist whose works exhibit many mathematical concepts, including tesselations, infinity, and polyhedra.
Related MiniText: Mathematical Art of M.C. Escher
Euclidean geometry
The theory of geometry arising from the five postulates (axioms) of Euclid: - That any two points determine a line;
- That any line may be extended indefinitely;
- That a circle may be drawn with any center and radius;
- That all right angles are equal to one another;
- That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.
Euclidean geometry is characterized by the 5th postulate, also called the “parallel’s postulate,” which may be restated as; given a line and a point not on the line, exactly one line may be drawn through the given point parallel to the given line. Theories of geometry which reject the 5th postulate form the subject of non-Euclidean geometry. Euclid recorded his theory of geometry in his 13-book opus titled Elements, c. 300 b.c., and his theory was considered the only “true” theory of geometry until the 18th century, when non-Euclidean geometry was discovered independently by Nikolai Lobachevsky, Janos Bolyai, and Karl Friedrich Gauss. Euclid’s axioms were found to be deficient in the 19th century, and Euclidean geometry was re-axiomatized by David Hilbert with the addition of a between-ness axiom, and the relegation of the concepts of point and line, which Euclid had defined, to the status of undefined terms.
Euler number
The transcendental number e, approximately 2.71828.... It may be defined as the limit
or as the infinite sum
The function ex is called the exponential function, and has the property that it is its own derivative.
even function
A real-valued function y = f(x) is even if f(x) = f(–x) for all x in the domain of f. The graph of an even function in the Cartesian plane is symmetric with respect to the y-axis.
Cf. odd function.
existential quantifier
See predicate calculus.
exp
See exponential function.
exponent
Also called index, a number or expression applying to another number or expression, called the base, and indicating that the base is to be “raised to the power of” the exponent. For integers, this corresponds to the operation of repeated self-multiplication the number of times specified by the exponent.
The exponent is written to the right of, and superscripted to, the base.
Cf. rational exponent, laws of exponents.
exponential function
The function ex. We say that an expression is expontiated when it is used as an exponent on e.
More generally, any function which operates on its argument by placing it as an exponent may be called an exponential function. Example: f(x) = 2x.
Cf. Euler number.
extended real numbers
The set of real numbers together with a “point(s) at infinity.” If both a positive and a negative infinity are added, then the resulting space is order equivalent and topologically equivalent (i.e., homeomorphic) to the unit interval [0, 1]. If only a single point at infinity is added, the resulting space is topologically equivalent to the unit circle.
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