By Singer I.

ISBN-10: 0471160156

ISBN-13: 9780471160151

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**Example text**

X) = g {J(X)) c g(Y) and hence g is surjective. Further, if x 1 E X and x2 E X, then (x 1 =f x2) ::::} (ex (xl ) =f ex (x2)) ::::} ( (g o f) (x t ) =f (g o f) (x2)) ::::} ::::} ( g {J (xl ) ) ) =f g (J (x2)) ::::} ( J (x 1 ) =f f (x2)) , and therefore f is injective. 0 Using the operation of composition of mappings one can describe mutually inverse mappings. Proposition. The mappings f : X � Y and g : Y � X are bijective and mutually inverse to each other if and only if g o f = ex and f o g = ey .

F(x) } consisting of the elements x E X that do not belong to the set f(x) E P(X) assigned to them by the bijection. Since A E P(X) , there exists a E X such that f ( a ) = A. For the element a the relation a E A is impossible by the definition of A, and the relation a fj. A is impossible, also by the definition of A. We have thus reached a contradiction with the law of excluded middle. 0 < < < < < : 1 8 R. Dedekind (1831-1916) - German algebraist who took an active part in the development of the theory of a real number.

It follows from the proposition just proved that we could have given a less intuitive, yet more symmetric definition of mutually inverse mappings as those mappings that satisfy the two conditions g o f = ex and f o g = ey . 4 Functions as Relations. The Graph of a Function In conclusion we return once again to the concept of a function. We note that it has undergone a lengthy and rather complicated evolution. The term function first appeared in the years from 1673 to 1692 in works of G. Leibniz (in a somewhat narrower sense, to be sure) .

### Abstract convex analysis by Singer I.

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