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Function as a relation A subset R of the cartesian product X × Y is called a binary relation defined on elements of sets X and Y . Definition Let X and Y be any sets. A relation f which satisfies the following two conditions: ∀ x ∈ X ∃ y ∈ Y ( x, y ) ∈ f ; ∀ x ∈ X ∀ y 1 , y 2 ∈ Y [( x, y 1 ) ∈ f ∧ ( x, y 2 ) ∈ f ] ⇒ y 1 = y 2 ; is called a function (or mapping) from X to Y . Definition If a function f : X → Y takes on each value in its codomain ( Y = R f ) , f is called surjective (surjection, onto function): f : X → onto Y ⇔ df ∀ y ∈ Y ∃ x ∈ X y = f ( x ) . Definition If a function f : X → Y sends distinct elements of X to distinct elements of Y , f is called injective (injection, one-to-one function): f : X → 1 − 1 Y ⇔ df ∀ x 1 , x 2 ∈ X [ x 1 6 = x 2 ⇒ f ( x 1 ) 6 = f ( x 2 )] . By the contrapositive: f : X → 1 − 1 Y ⇔ df ∀ x 1 , x 2 ∈ X [ f ( x 1 ) = f ( x 2 ) ⇒ x 1 = x 2 ] . Definition If a function f is both injective and surjective, then f is called bijective (bijection, one-to-one onto function). If f is bijective, then there exists its inverse function f − 1 : Y → X , such that: ∀ x ∈ X ∀ y ∈ Y [ x = f − 1 ( y ) ⇔ y = f ( x )] .
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