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relation – Sid’s Paradox
relation
An n-place relation is defined on a Cartesian product of n sets, and is represented by a set of ordered n-tuples. For example, the less-than (“<”) relation is a binary relation on numbers, and the membership relation (“e”) is a binary relation on sets. The property of forming a Pythagorean triple is a ternary relation on natural numbers, of which for example (3,4,5) is a member since 32 + 42 = 52.
 In a binary (two-place) relation, the set from which the abscissae are taken is called the domain, and the set providing the ordinates is called the range. Binary relations are classified according to whether they are reflexive, transitive, and/or symmetric.
Cf. function, partial order, lattice.
relatively large
A set A of natural numbers is called relatively large if the number of elements of A is greater than the least element of A.
relatively prime
Two natural numbers a and b are relatively prime if their greatest common divisor is 1.
right angle
An angle of 90 degrees (p/2 radians). Equivalently, it can be said that two right angles are supplemental angles, i.e., they add up to a straight line (180 degrees or p radians).
Cf. complementary angles, acute, obtuse.
ring of sets
Given a set X, a ring on X is a collection of subsets of X which is closed under finite unions and set differences. If the ring includes X itself then it is an algebra of sets. If the ring is closed under countable unions, then it is called a s-ring.
root
An nth root of a real or complex number x is a number which when multiplied by itself n times yields x.
Of a polynomial p: A number x such that p(x) = 0.
Russell Paradox
(Bertrand Russell, 1901) A paradox of set theory which necessitated a more careful axiomatization of set theory in the 1920’s and 1930’s: Naively, some sets are members of themselves and some are not. For instance, the set of all apples is not itself an apple, but the set of all sets does seem to be a set. So consider the set X of all sets that are not members of themselves. We may ask, is X a member of itself? If it is then it cannot be, because of the way in which X itself was defined, but if it isn’t then it must be, by the same reasoning. Contradiction. The Russell paradox is resolved in modern set theory by a foundation axiom or axiom of regularity, and by limiting the “size” of objects we call sets. For example, the “set of all sets” is considered not to be a set but a proper class.
scalar
A quantity having only magnitude, not direction (typically an element of a field, such as the real numbers or complex numbers).
Cf. vector.
scalene
A triangle is called scalene if all of its sides are unequal (equivalently, if all of its angles are unequal).
Schroeder-Bernstein Theorem
If there exists an injection from a set X into a set Y, and also an injection from Y into X, then there exists a bijection from X to Y, and hence X and Y have the same cardinality.
scientific notation
A number is written in scientific notation when it is written as the product of a real number between 1 and 10 and a power of 10. E.g., 320 is written in scientific notation as 3.2 × 102. On some calculators and in some textbooks, this may be written as 3.2E2. Scientific notation is a convenient way to represent very large and very small numbers.
semi-lattice
A set with a single binary operation that is idempotent, commutative, and associative.
Cf. lattice.
sentential calculus
See propositional calculus.
sequence
A sequence is a set (of numbers, or sets, or functions, etc.) indexed by the natural numbers. Sequences may be infinite, and may be regarded as a function with domain the set of natural numbers and range the set of objects in the sequence.
An infinite sequence of numbers is said to converge to a number L provided that, given any positive e, we may find a natural number N such that for all terms of the sequence after the N th one, their difference from L is less than e. Naively, the terms of the sequence eventually become “arbitrarily close” to L. Such a sequence is called convergent, and the number L is called the limit of the sequence, or the limit point, or sometimes the accumulation point of the sequence.
 Alternatively, a cluster point or accumulation point P of a sequence may be defined as a point with the property that infinitely many terms of the sequence lie in any neighborhood of P. A sequence may have more than one such cluster point (even infinitely many).
A sequence is called Cauchy if, for every e greater than zero, we may find a natural number N so that the difference between any two terms following the N th term is smaller than e. Every convergent sequence is Cauchy; the converse is true in complete spaces.
Cf. series.
Related article: Limits
set
Naively, any well-defined collection considered as a single, abstract object. By “well-defined” is meant that it is always possible to determine for a given set when something is an element of the set and when not. In formal set theory, the term “set” is not defined, but is a primitive term whose meaning is informed purely by the axioms in which it appears.
Cf. ZF, ZFC.
set algebra
See algebra of sets.
set difference
The set difference of two sets A and B, denoted A - B or A \ B, is the set of elements that is contained in A but not in B. This is equivalent to the intersection between A and the complement of B.
set function
A function whose domain of definition is a collection of sets.
set ring
See ring of sets.
set theory
Naive set theory: The study of sets (i.e., well-defined collections of objects) which have a binary extensional relation (set membership) defined on them.
Abstract set theory: As naive set theory, but with all sets built using only elements which are themselves sets (beginning with the empty set, which has no members).
Formal set theory: Any of several axiom systems of abstract set theory in the language of first-order logic, such as Zermelo Fraenkel set theory, Gödel-Bernays set theory, Quine’s New Foundations, etc.
Sid’s Paradox
ARTICLE
A paradox of knowledge: is the distinction between matters of opinion and matters of fact a matter of opinion or a matter of fact? See the article for discussion.
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