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jectiva
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There are 7 letters in JECTIVA ( A1C3E1I1J8T1V4 )
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In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other. * A function maps elements from its domain to elements in its codomain. Given a function * * * * f * : * * X * → * Y * * * {\displaystyle f:\;X\to Y} * * The function is injective (one-to-one) if each element of the codomain is mapped to by at most one element of the domain. An injective function is an injection. Notationally: * * * * ∀ * x * , * * x * ′ * * ∈ * X * , * f * ( * x * ) * = * f * ( * * x * ′ * * ) * ⇒ * x * = * * x * ′ * * . * * * {\displaystyle \forall x,x'\in X,f(x)=f(x')\Rightarrow x=x'.} * * Or, equivalently (using logical transposition), * * * * * ∀ * x * , * * x * ′ * * ∈ * X * , * x * ≠ * * x * ′ * * ⇒ * f * ( * x * ) * ≠ * f * ( * * x * ′ * * ) * . * * * {\displaystyle \forall x,x'\in X,x\neq x'\Rightarrow f(x)\neq f(x').} * The function is surjective (onto) if each element of the codomain is mapped to by at least one element of the domain. (That is, the image and the codomain of the function are equal.) A surjective function is a surjection. Notationally: * * * * ∀ * y * ∈ * Y * , * ∃ * x * ∈ * X * * such that * * y * = * f * ( * x * ) * . * * * {\displaystyle \forall y\in Y,\exists x\in X{\text{ such that }}y=f(x).} * The function is bijective (one-to-one and onto or one-to-one correspondence) if each element of the codomain is mapped to by exactly one element of the domain. (That is, the function is both injective and surjective.) A bijective function is a bijection.An injective function need not be surjective (not all elements of the codomain may be associated with arguments), and a surjective function need not be injective (some images may be associated with more than one argument). The four possible combinations of injective and surjective features are illustrated in the diagrams to the right. |