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subset – Tarski Truth Theorem
subset
A set A is a subset of a set B if every element of A is also an element of B. If in addition B is a subset of A, then A = B, but if not then A may be said to be a proper subset of B.
 Cf. superset.
Substitution Principle
Intuitively, the assertion that if one “plugs” a value into a formal polynomial, everything will work out as expected.
Formally, the theorem that, given two commutative rings R and R', a homomorphism f from R to R', and an element r of R', there is a unique homomorphism F from the polynomial ring R[x] over R to R', with the properties - F(a) = f(a) for all a in R,
- F(x) = r.
For any polynomial p, Fr(p) is written p(r), and corresponds to “evaluating p on r.”
subtract
To subtract a number m from a number n is to calculate the difference of m and n. If m is less than n we take the positive difference, otherwise we take the negative of the difference. This is tantamount to adding the negative of m to n.
successor
In a structure with an order relation defined upon it, the successor of an element a is the least element greater than a, if such exists.
Cf. predecessor.
sumset
Given a set A, the sumset of A, denoted by
is the set containing all of the elements of the elements of A, that is, it is the union of the elements of A.
sumset axiom
An axiom of set theory which states that if A is any set, then the sumset of A is also a set.
sup
Abbreviation of supremum.
superset
A set A is a superset of a set B if every element of B is an element of A.
Cf. subset.
supplemental angles
Two angles are supplemental if they add up to 180 degrees (p radians).
Cf. complementary angles.
supremum
The supremum of any subset of a linearly ordered set is the least upper bound of the subset. In particular, the supremum of any set of numbers is the smallest number in the set which is greater than or equal to every number in the set. In a complete linear order the supremum of any bounded set always exists.
Cf. infimum, least upper bound axiom.
surd
(rare) An irrational root of a number, e.g., the square root of two.
Related article: Irrationality of the Square Root of 2
surjection
A surjective function, i.e., a function that maps at least one element of its domain to each element of its range.
Cf. injection, bijection.
surjective
A function f from a set X to a set Y is surjective, also called “onto,” if to each element y of Y there is an element x of X such that f maps x to y, i.e., f (x)=y. Compare: injective, bijective.
Suslin tree
Set Theory: For a an infinite cardinal, an a-Suslin tree is a tree T such that |T| = a, and every chain and every antichain of T has cardinality less than a.
Cf. Aronszajn tree.
symbolic logic
Logic reduced to syntax, i.e., which works only with uninterpreted symbols. The two most often used kinds of symbolic logic are the propositional calculus and the predicate calculus.
symmetric difference
The symmetric difference of two sets A and B is the set of those elements that are in either A or B but not both.
Cf. intersection, union.
symmetric group
The symmetric group on a set is the group of permutations of its elements. On a set with n elements, there are n! permutations.
Cf. alternating group.
symmetric relation
A relation “ ~ ” on a set X is symmetric if for any two elements x and y in X we have x ~ y if and only if y ~ x. The relation “ ~ ” is called asymmetric if for any two elements x and y we have that x ~ y implies it is not true that y ~ x, and it is called antisymmetric if whenever x ~ y and y ~ x then x = y. Note that a relation may be neither symmetric, asymmetric, nor antisymmetric.
Cf. reflexive relation, transitive relation.
tan
See tangent.
tangent
Geometry: A tangent to a curve at a point P is the straight line through P whose slope is equal to the slope of the curve at P.
Trigonometry: A periodic trigonometric function defined on angles, usually abbreviated “tan.” Can be defined independently or in terms of the trig functions sine and cosine.
See the related article for a complete exposition.
Related article: Trig Functions and Identities
Tarski Truth Theorem
Let T be a mathematical theory. Denote formulas of T by j, y, etc., and let T j be the statement that “the sentence j is true in T ” Choose a canonical numbering of the formulas of T (e.g., Gödel numbering) that assigns to each sentence of T a unique positive integer, and denote the positive integer associated with a sentence j by ‘j’. Then there is no formula y in T such that y(‘j’) Tj.
In other words, “truth” in a theory T is not definable in T.
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