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ordinal – Platonic solid
ordinal
The class of ordinals is defined by:- 0 is an ordinal;
- if a is an ordinal, then a + 1 = a union {a}) is an ordinal;
- if A is a collection of ordinals, then union(A) is an ordinal;
- nothing else is an ordinal.
The class of ordinals is transitive, and is a well-founded, linear ordering. An ordinal of the form a + 1 is called a successor ordinal, and is otherwise called a limit ordinal.
 Cf. Von Neumann Heirarchy.
ordinate
The second element of an ordered pair.
Cf. abscissa.
outer measure
A non-negative extended real-valued set function defined on all subsets of a space X that is zero on the empty set, monotonic, and countably subadditive (see below) is called an outer measure. An outer measure is often used together with Caratheodory's Theorem (see below) to obtain a measure. Given a set X and a collection A of subsets of X which includes the empty set and X itself, and a positive real-valued function r whose domain is A and whose value on the empty set is zero, then for any F in X define m* by
Then m* is an outer measure. A set B in X is then called m*-measurable if
for all subsets C of X. (Cc denotes the complement of C in X.) Carathéodory’s Theorem states that if m* is an outer measure on X, then the collection M of m*-measurable sets is a s-algebra of sets, and the restriction of m* to M is a complete measure.
parabola
The locus of points in the plane whose distances from a fixed point, called the focus, and a fixed line, called the directrix, are equal.
Like the ellipse and hyperbola, the parabola is a conic section. See the related article for a full exposition.
Related article: Conics
paradox
A seemingly 'necessary' contradiction or absurdity. A paradox arising logically out of formal axioms is called an antinomy. True paradoxes may be broadly classified as paradoxes of logic, paradoxes of infinity, paradoxes of knowledge, paradoxes of language, and paradoxes of self-reference. See:Antistrephon Paradox
Banach-Tarski Paradox
Berry Paradox
Boundary Paradox
Finitude Paradox
First Boring Number Paradox
Grelling's Paradox
Grue-Bleen Paradox
Liar Paradox
Quine's Paradox
Prisoner's Dilemma
Russell Paradox
Santa Sentence Paradox
Sid's Paradox
Sorites Paradox
Unexpected Hanging Paradox
Zeno's Paradox of the Arrow
Zenos Paradox of the Moving Rows
Zeno's Paradox of the Tortoise and Achilles
Paradox of the Heap
See Sorites Paradox.
parallelogram
A quadrilateral with opposite sides (and opposite angles) equal.
parallel postulate
The 5th postulate of Euclidean geometry.
partially ordered set
A set with a partial order defined on it.
partial order
A partial order on a set X is a binary relation “ ” on X that is reflexive, transitive, and antisymmetric. A partial order in which every pair of elements is related is called a total order or linear order. A partial order on X imposes a lattice on X.
Cf. linear order, tree.
partition
General: A partition of a set X is a collection of subsets of X such that every element of X is in exactly one of the subsets. Such a partition is given by (and gives rise to) an equivalence relation on X. For example, division modulo 3 partitions the set of natural numbers into three subsets, each containing all those numbers leaving remainders of 0, 1, or 2 respectively when divided by 3.
Algebra: A partition of a matrix is a division of the matrix into conformable submatrices.
Analysis: A partition of a space is a collection of pairwise disjoint regions of the space whose union is the entire space. For example, a partition of an interval [a, b] of the real line is given by a finite set of points {xi} such that a = x1 < x2 < . . . < xn = b which divide the interval into disjoint subintervals.
Number Theory: Given a positive integer n, a partition of n is a set of positive integers whose sum is n. For example, 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1 are the four possible partitions of the number 4.
Peano, Giuseppe
(born 1858)
Italian mathematician best known for the Peano axioms, which define the natural numbers in terms of sets and formalize arithmetic in a logically rigorous way. Peano was primarily an analyst, however, and was the inventor of space-filling curves, that is surjective functions from the unit interval to the unit square.
Peano axioms
The axiom system developed by Giuseppe Peano to formalize arithmetic. The key to his method is the introduction of a “successor operation” S on numbers; if a is a number, then Sa is the successor of that number. In this way Peano reduced arithmetic to the conceptually primitive operation of counting.
The axioms are:- 0 is a number.
- If a is a number, then Sa is a number.
- If a and b are numbers, then a + 0 = a, and a + Sb = S(a + b).
- If a and b are numbers, then a × 0 = 0, and a × Sb = a × b + a.
- (Induction Principle.) For any formula f, if we have f(0) and if f(a) always implies f(Sa), then we have f(a) for all numbers a.
perfect number
A natural number n whose distinct divisors, including 1 but not including n itself, sum to n.
Example: 6 is perfect since its divisors are 1, 2, and 3, and 1 + 2 + 3 = 6. The first three perfect numbers are 6, 28, and 496. Any number of the form 2n – 1(2n – 1) is perfect provided that 2n – 1 is a Mersenne prime. Such numbers are called Euclid numbers. Euler proved that all even perfect numbers are Euclid numbers. It is not known whether there are infinitely many perfect numbers, or if there are any odd perfect numbers.
perfect set
A closed set X is called perfect if every point of X is an accumulation point of X. The following are equivalent characterizations:- X is closed and containes no isolated points.
- X is closed and dense in itself.
- X is equal to its derived set.
permutation
A permutation on a finite set is a reordering of its elements. On a set of n elements there are n! (n-factorial) permutations, and together they form a group, the symmetric group on n elements.
The number of permutations of a subselection of r elements from a set of n elements is usually denoted nPr, and is given by
A cyclic permutation, also called a cycle, of an ordered collection of n objects is obtained by replacing each kth object with the (k - 1)th object, and replacing the first object with the nth object. In other words, each object advances one place in the order, and the last object “wraps around” to the first position. The number of objects n is called the degree (sometimes length) of the cycle. A cycle of length 2 is called a transposition. Every permutation can be factored as a product of transpositions, and is called an even (or odd) permutation if it factors into an even (resp. odd) number of transpositions.
The subgroup of the symmetric group on n elements consisting of all the even permutations is called the alternating group on n elements.
Cf. combination.
perpendicular
Two lines are perpendicular if they intersect in a right angle. In particular, two lines in the Cartesian plane are perpendicular if their respective slopes m1 and m2 satisfy m 1 × m2 = –1.
Two curves intersect perpendicularly if their tangent lines at the point of intersection are perpendicular.
Perpendicularity of planes and surfaces is defined analogously. Cf. normal.
perpendicular distance
Geometry: The shortest distance between a point and a line, or between a point and a plane. Sometimes called directed distance.
Pi (p)
The real number that is equal to the ratio of the circumference of any circle to its diameter, approximately equal to 3.1415926. It is known to be a transcendental number.
plane figure
A figure, i.e., a graphical (visual) representation of points, lines, curves, or regions in the geometric plane. A plane figure, such as a polygon, that bounds a finite region is called closed.
Platonic solid
ARTICLE
A solid having congruent, regular polygonal faces. There are five Platonic solids: the icosahedron, tetrahedron, octahedron, dodecahedron, and cube.
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