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power set – Ramsey’s Theorem
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.
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.
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
Ramsey cardinal
A cardinal k is called a Ramsey cardinal if
Cf. Ramsey’s Theorem.
Ramsey number
See Ramsey’s Theorem.
Ramsey’s Theorem
Graph theory: given any two positive integers m and n, there exists a number, called the Ramsey number of m and n, and denoted R(m, n), such that any simple graph with R(m, n) vertices either contains a clique with m vertices (all vertices adjacent) or an independent set with n vertices (no vertices adjacent).
The question of how to determine the Ramsey number for arbitrary m and n is unsolved and believed to be very difficult, but various lower and upper bounds on it have been proven over the years. In particular, it is known that R(m, m)  2m/2, and that R(m, n)  (m + n - 2)C(m - 1).
Also, certain specific Ramsey numbers are known, e.g., R(3, 3) = 6. Thus, in any group of six people, either three are mutual acquaintances or three are mutual strangers.
Set theory: Let n be a positive integer, k and l be (finite or transfinite) cardinals, and denote by k ln the assertion that the complete graph of size k in which each edge has been “colored” by one of n colors contains a complete subgraph of size l all of whose edges are the same color. More generally still, denote by
the assertion that if the subsets of k of size b have each been colored with one of a-many colors, then there is a subset of X of size l all of whose b-sized subsets are the same color (homogeneous). Ramsey’s Theorem (infinite case) may now be stated as: For all positive integers k and l, we have
i.e., for every coloring of the k-sized subsets of the natural numbers (w) into l colors, there is an infinite homogeneous set (an infinite collection of k-sized subsets all the same color).
This more general notion gives rise to a field of set theory known as Ramsey Theory, in which far more is known about infinite sets than finite ones.
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