Algèbre relationnelle : Fonctions, Relations et Ensembles — Terminale Math

Question 1 sur 10 10:00

[{"id":10283,"question":"Soit f : E → F une fonction. Si pour tout y ∈ F, il existe un unique x ∈ E tel que f(x) = y, alors f est :","option_a":"Injective","option_b":"Surjective","option_c":"Bjective","option_d":"Non définie","option_e":"","option_f":"","bonne_reponse":"C","explication":"Une fonction est bijective si elle est à la fois injective (chaque élément de E a une image unique) et surjective (tout élément de F est atteint).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10284,"question":"L'ensemble {1, 2, 3} × {a, b} contient exactement :","option_a":"5 éléments","option_b":"6 éléments","option_c":"9 éléments","option_d":"12 éléments","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le produit cartésien de deux ensembles de cardinaux 3 et 2 contient 3 × 2 = 6 éléments.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10285,"question":"Une relation R sur un ensemble E est réflexive si :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Une relation R est réflexive si pour tout x ∈ E, (x, x) ∈ R.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10286,"question":"Soit f : ℝ → ℝ définie par f(x) = x². Cette fonction est :","option_a":"Injective","option_b":"Surjective","option_c":"Ni injective ni surjective","option_d":"Bjective","option_e":"","option_f":"","bonne_reponse":"C","explication":"f n'est pas injective car f(2) = f(-2) = 4, et elle n'est pas surjective car les réels négatifs n'ont pas d'antécédent.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10287,"question":"Le cardinal de l'ensemble P({1, 2}) (parties de {1, 2}) est :","option_a":"2","option_b":"3","option_c":"4","option_d":"8","option_e":"","option_f":"","bonne_reponse":"C","explication":"L'ensemble {1, 2} a 2 éléments, donc son ensemble de parties a 2² = 4 éléments : ∅, {1}, {2}, {1, 2}.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10288,"question":"Soit f : E → F une fonction. Si f est injective, alors :","option_a":"|E| ≤ |F|","option_b":"|E| ≥ |F|","option_c":"|E| = |F|","option_d":"Aucune des réponses","option_e":"","option_f":"","bonne_reponse":"A","explication":"Une fonction injective associe à des éléments distincts de E des images distinctes dans F, donc |E| ≤ |F|.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10289,"question":"L'ensemble {x ∈ ℝ | x² = 4} est égal à :","option_a":"{2}","option_b":"{-2}","option_c":"{2, -2}","option_d":"∅","option_e":"","option_f":"","bonne_reponse":"C","explication":"Les solutions réelles de x² = 4 sont x = 2 et x = -2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10290,"question":"Une relation d'équivalence est toujours :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Une relation d'équivalence est réflexive, symétrique et transitive, mais pas nécessairement une fonction (elle peut associer plusieurs éléments à un même élément).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10291,"question":"Soit f : ℕ → ℕ définie par f(n) = n + 1. Cette fonction est :","option_a":"Injective mais pas surjective","option_b":"Surjective mais pas injective","option_c":"Bjective","option_d":"Ni injective ni surjective","option_e":"","option_f":"","bonne_reponse":"A","explication":"f est injective car f(n1) = f(n2) ⇒ n1 = n2, mais elle n'est pas surjective car 0 n'a pas d'antécédent.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":10292,"question":"Le produit cartésien de deux ensembles infinis dénombrables est :","option_a":"Fini","option_b":"Dénombrable","option_c":"Non dénombrable","option_d":"Indéterminé","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le produit cartésien de deux ensembles dénombrables est dénombrable (théorème de Cantor).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]