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Intuitively, a function is a process that associates each element of a set X, to a single element of a set Y Formally, a function f from a set X to a set Y is defined by a set G of ordered pairs (x, y) such that x ∈ X, y ∈ Y, and every element of X is the first component of exactly one ordered pair in G In other words, for every x in X, there is exactly one element y such that the. C X g F Γc q L 炩 炨 ߂ ܂ B ` F W A N V q o f } W i q Y @ v C. The uniqueness can be proved as follows Let d = max(1 deg f, deg g)All together, b and the a ij have d coefficients The shape of the decomposition defines a linear map from coefficient vectors to polynomials f of degree less than dThe existence proof means that this map is surjectiveAs the two vector spaces have the same dimension, the map is also injective, which means uniqueness of.

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