BROWSE ALPHABETICALLY LEVEL:    Elementary    Advanced    Both INCLUDE TOPICS:    Basic Math    Algebra    Analysis    Biography    Calculus    Comp Sci    Discrete    Economics    Foundations    Geometry    Graph Thry    History    Number Thry    Phys Sci    Statistics    Topology    Trigonometry order relation – polyhedron order relation   A relation R on a set S is an order relation exactly if it is reflexive, transitive and antisymmetric. Order relations are usually denoted by “ < ” or “ ”.Cf. partial order, total order order type   See total order. 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. 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 ParadoxBanach-Tarski ParadoxBerry ParadoxBoundary ParadoxFinitude ParadoxFirst Boring Number ParadoxGrelling's ParadoxGrue-Bleen ParadoxLiar ParadoxQuine's ParadoxPrisoner's DilemmaRussell ParadoxSanta Sentence ParadoxSid's ParadoxSorites ParadoxUnexpected Hanging ParadoxZeno's Paradox of the ArrowZenos Paradox of the Moving RowsZeno's Paradox of the Tortoise and Achilles Paradox of the Heap   See Sorites Paradox. parallelogram   A quadrilateral with opposite sides (and opposite angles) equal. 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 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 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 byA 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. polygon   A closed plane figure having straight sides. A polygon that has equal sides and equal angles is called a regular polygon. Examples: square, hexagon, nonagon, etc. polyhedron   A closed, solid figure consisting of four or more planar faces, pairs of which meet at edges. The faces form polygons, and the points where three or more edges meet are called vertices. The number of faces, edges, and vertices of a polyhedron are related by the Euler polyhedron formula.A polyhedron with four faces is called a tetrahedron, a pentahedron has five faces, a hexadron six, a heptahedron seven, an octahedron eight, a dodecahedron twelve, and an icosahedron twenty. If all the faces of a polyhedron are congruent, it is called a regular solid or Platonic solid. order relation – polyhedron
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