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11. 28. I. 12. rtie de a. s à b. ('v'a) ('v'b Ca) (3Cab) ('v'x) [(x E Cab)+-+ (x E a) A (x '/. 29) Plus généralement, si a et b sont deux ensembles quelconques, il existe un ensemble unique noté a\ b (resp. s à b (resp. l à la. réunion des ensembles a\ b et b \a). 31) DÉMONSTRATION: - ri. b)) Posons Cab= {x E a: X rf. b}, alors l'axiome de sélection nous permet de dire que Cab est un ensemble et cet ensemble est unique d'après l'axiome d'extensionnalité. On fait le même raisonnement pour l'ensemble a\ b.

12) permet de dire qu'il existe t E An B tel que l(t) = z. Or t E {x} n {y} implique nécessairement t =X= y, donc I est bien injective. = = = = = = (*)Soit I une application constante, I (X) b quel que soit X E E. Soient A et B deux parties disjointes de E. An B = 0, donc l(A n B) 0 qui est distinct de l(A) n l(B) {b}. aire de A par f est encore = { b} qui est distinct du complémentaire de I (A) = {b}. = §3. 9). Supposons f injective, et soit x

On dit qu'on a effectué le changement de variable u, ou changement d'ensemble initial E 1 1-+ E, et que fi est l'image réciproque de f par ce changement de variable. On obtient l'expression de fi(x1), x1 E Ei, en faisant, dans l'expression de f(x) la substitution x = u(xi). ) , représentée à l'aide de la variable x 1 au lieu de la variable x. 12. - Soit f une fonction définie sur E, à valeurs dans F. Si v est une application de F dans un ensemble F 2 , on peut définir une nouvelle fonction '2 = v o f définie sur E à valeurs dans F 2 .

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