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Peano axioms – power set axiom
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.
pointwise bounded
See bounded.
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.
polynomial
General: A function of the form
p(x) = a0 + a1x + a2x2 + ... + anxn
where the ai (called coefficients) are real numbers or complex numbers, and the exponents are all natural numbers. The highest exponent is called the degree of the polynomial, and the coefficient an on the highest degree term is called the leading coefficient. Polynomials of degree two are called quadratic polynomials, of degree 3 cubic, of degree 4 quartic, and those of degree 5 are called quintic.
More generally, a polynomial may be in several variables x1, ... ,xk, and may be thought of as a sum of the form
where all but finitely many of the ai are 0. In this case, the degree of the polynomial is the highest sum of exponents appearing in any term. For example, 2x4 and 3xy2z are both fourth-degree terms.
Abstract Algebra (formal polynomial): A formal sum of the form
The ai are called coefficients, and are elements of some commutative ring R. p is said to be a polynomial over R, or with coefficients in R. The x is just a formal symbol. Only finitely many of the ai can be non-zero, and if an is the last non-zero coefficient, n is called the degree of p.
Polynomials can be added and multiplied in the natural way, by using the cummutative and distributive laws of addition and multiplication in R. Explicitly, if p and q are polynomials over the same ring, and the ai and bi are the coefficients of p and q, respectively, then
and
These operations allow us to define R[x], the polynomial ring over R.
The Substitution Principle allows us evaluate a polynomial p on any element of R, and we can use this to define a function corresponding to p, thereby capturing the informal notion of a polynomial.
Cf. Diophantine equation, quadratic formula.
polynomial ring
The ring, denoted by R[x], of all formal polynomials over a given commutative ring R. If we choose to consider multivariate polynomials, we can define the ring R[x1, x2, ... xn] in an analagous manner. A worthwhile observation, which is a corollary of the Substitution Principle, is that R[x1, x2, ... xn] is isomorphic to R[x1][x2] ... [xn].
poset
A partially ordered set, i.e., a set with a partial order defined on it.
positive set
Given a signed measure m on a measure space X, a measurable set A in X is called a positive set if the measure of all measurable subsets of A is greater than or equal to zero.
Cf. negative set, null set.
postulate
A statement in a mathematical theory that is assumed without proof. Essentially synonymous with “axiom.”
potential infinite
A distinction made by Aristotle: a set is potentially infinite if it cannot be finitely completed, e.g., our naive or given conception of the natural numbers. Aristotle admitted potentially infinite sets, but denied the logical possiblity of the actual infinite, that is, infinite totalities considered as completed entities.
Related MiniText: Infinity -- You Can't Get There From Here...
power series
An infinite series of the form
See the related article for a complete description.
Related article: Series
power set
Given a set X, the power set of X, denoted P(X), is the collection of all subsets of X. If X is a finite set with n elements, then P(X) is a finite set with 2n elements.
Cf. power set axiom.
power set axiom
An axiom of set theory which states that, for any given set X, the power set, i.e., the collection of all subsets of X, exists and is a set.
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