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This disagreement is confusing, but we're stuck with it. Some authors use "one-to-one" as a synonym for "injective" rather than "bijective". If A 1, A 2, A 3, A 4 and A 5 were Assistants C 1, C 2, C 3, C 4 were Clerks M 1, M 2, M 3 were managers and E 1, E 2 were Executive officers and if the relation R is defined by xRy, where x is the salary given to person y, express the relation R through an ordered pair and an arrow diagram. Which must also be bijective, and therefore onto. The onto function from Y to X is F's inverse. Here are some equivalent ways of saying that T is onto: The range of T is equal to the codomain of T.
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Therefore, by definition a one-to-one function is both into and onto.īut you say "an onto function from Y to X must exist." The "from Y to X" part might be what's tripping you up? F is onto, but it's from X to Y. Definition(Onto transformations) A transformation T : R n R m is onto if, for every vector b in R m, the equation T ( x ) b has at least one solution x in R n. It is onto (aka surjective) if every element of Y has some element of X that maps to it:Īnd for F to be one-to-one (aka bijective), both of these things must be true. f is bijective if it is surjective and injective (one-to-one and onto). ∀ x ∈ X, ∃ y ∈ Y | f(x) = y x 1 ≠ x 2 ⇒ f(x 1) ≠ f(x 2) In this case, the range of f is equal to the codomain. I If f maps element a 2 A to element b 2 B, we write f. I A is calleddomainof f, and B is calledcodomainof f. A function F: X → Y is into (aka injective) if every element of X is mapped to a distinct element of Y: CS311H: Discrete Mathematics Functions Instructor: Is l Dillig Instructor: Is l Dillig, CS311H: Discrete Mathematics Functions 1/46 Functions I Afunction f from a set A to a set B assigns each element of A to exactly one element of B.