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Additionally, a family of sets may be defined as a function from a set , known as the index set, to , in which case the sets of the family are indexed by members of . [1] In some contexts, a family of sets may be allowed to contain repeated copies of any given member, [ 2 ] [ 3 ] [ 4 ] and in other contexts it may form a proper class .
Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer coefficients.. Polynomials: Can be generated solely by addition, multiplication, and raising to the power of a positive integer.
It is the fact that many mathematical theories (even the "purest") have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced. [126]
This set-theoretic definition is based on the fact that a function establishes a relation between the elements of the domain and some (possibly all) elements of the codomain. Mathematically, a binary relation between two sets X and Y is a subset of the set of all ordered pairs ( x , y ) {\displaystyle (x,y)} such that x ∈ X {\displaystyle x ...
A – adele ring or algebraic numbers. a.a.s. – asymptotically almost surely. AC – Axiom of Choice, [1] or set of absolutely continuous functions. a.c. – absolutely continuous.
Dudley Weldon Woodard (October 3, 1881 – July 1, 1965) was a Galveston-born American mathematician and professor, and the second African-American to earn a PhD in mathematics; the first was Woodard's mentor Elbert Frank Cox, who earned a PhD from Cornell in 1925).