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power set – radian
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.
precompact
Given a topological space X, a subset E of X is called precompact if its closure is compact.
predecessor
In a structure with an order relation defined upon it, the predecessor of an element b is the greatest element less than b.
Cf. successor
predicate calculus
A system of symbolic logic which augments the propositional calculus with quantification over variables. The two forms of quantification are existential and universal, and are denoted by
respectively. This permits the construction of sentences such as
which could be read, “There exists an x such that for all y, x times y is equal to y.” (Such a sentence would be true in arithmetic or group theory, for instance.) When quantification is permitted only over variables, the logic is first-order. If quantification is permitted over classes of variables or over predicates, the logic is second-order.
prime number
Any natural number greater than 1 that is evenly divisible only by itself and 1. There are infinitely many prime numbers. The number of primes less than a given number n is denoted p(n), and approaches the value n/lnn for sufficiently large n.
Related article: Fundamental Theorem of Arithmetic
product
The result of applying a multiplication operation to two or more numbers or quantities.
proper class
A collection of elements that is not a set. For example, the collection of all sets must be thought of as a proper class in order to avoid the Russell paradox.
proper factor
See factor.
propositional calculus
The formal system of symbolic logic in which sentences are treated as objects related by the logical connectives:
The last two symbols are called the conditional and the biconditional respectively, but they are not essential; indeed all the connectives are fully expressible by the use of the two connectives for “or” and “not.” For example, “p implies q” may be equivalently expressed as “not p or q.” In the notation of propositional calculus, this equivalence may be written,
Cf. predicate calculus.
p-series
An infinite series of the form
with p a positive real number. See the related article for details.
Related article: Series
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Pythagoras
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Pythagoras
(born 580 bce)
What we know of this ancient Greek mystic is mostly legend, for of the true facts of his life almost nothing is known. However, it is believed that he started a religious cult, the Pythagoreans, among whose beliefs was that number was the highest reality, and that all other aspects of reality could be understood in terms of integers and ratios of integers.
Pythagorean theorem
In Euclidean geometry, the sum of the areas of the squares on the legs of any right triangle is equal to the area of the square on the hypotenuse. This is arguably the most important theorem of classical mathematics, and perhaps of all time.
Pythagorean triple
An ordered triple (a,b,c) of natural numbers satisfying a2 + b2 = c2. The triples (3,4,5) and (5,12,13) are the first of infinitely many examples.
quadratic formula
Given a quadratic function, i.e., a polynomial function of second degree y = ax 2 + bx + c, the zeros of the function are given by
The expression under the radical is called the determinant. If the determinant is positive, both solutions are real; if negative, both solutions are complex; and if zero, there is a single solution of multiplicity two.
quadrilateral
A closed, plane figure with four straight sides.
Cf. polygon.
quantifier
See predicate calculus.
quasi-disjoint family
Set Theory: See D-system.
Quine atom
In set theory, a set whose only member is itself, i.e., x = { x }. More generally, some phrases may be “Quined” to form meaningful sentences, e.g., “is a five-word phrase” is a five-word phrase.
quotient
The number that results from dividing one number by another.
Cf. division algorithm.
radian
A dimensionless unit of measure of angles. An angle of one radian is given by the central angle of a circle subtending an arc of length equal to the radius of the circle. Consequently, 360 degrees is the same as 2p radians. See the related article for a more extensive exposition.
Related article: Trig Functions and Identities
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