Quiz — Applications et parties d’un ensemble, I.pdf

Question 1 sur 10 20:00

[{"id":22359,"question":"Soit E = {1, 2, 3}. Combien de parties E possède-t-il ?","option_a":"3","option_b":"6","option_c":"8","option_d":"12","option_e":"","option_f":"","bonne_reponse":"c","explication":"Un ensemble à n éléments possède 2ⁿ parties. Ici, n=3, donc 2³ = 8 parties.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"3\", \"b\": \"6\", \"c\": \"8\", \"d\": \"12\"}}","_debug_options_count":4},{"id":22360,"question":"Si A = {a, b} et B = {b, c}, que vaut A ∩ B ?","option_a":"{a, b, c}","option_b":"{b}","option_c":"{a, c}","option_d":"∅","option_e":"","option_f":"","bonne_reponse":"b","explication":"L’intersection A ∩ B est l’ensemble des éléments communs à A et B, ici {b}.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"{a, b, c}\", \"b\": \"{b}\", \"c\": \"{a, c}\", \"d\": \"∅\"}}","_debug_options_count":4},{"id":22361,"question":"L’ensemble vide est-il une partie de tout ensemble ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L’ensemble vide (∅) est toujours une partie de tout ensemble, selon la définition mathématique.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":22362,"question":"Soit f une application de E vers F. Que signifie f est injective ?","option_a":"Chaque élément de E a une image unique dans F","option_b":"Chaque élément de F a un unique antécédent dans E","option_c":"f est bijective","option_d":"f n’est pas définie sur tout E","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une application est injective si deux éléments distincts de E ont toujours des images distinctes dans F.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Chaque élément de E a une image unique dans F\", \"b\": \"Chaque é","_debug_options_count":4},{"id":22363,"question":"L’ensemble {1, 2, 3} est-il une partie de {1, 2, 3, 4} ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Tous les éléments de {1, 2, 3} appartiennent à {1, 2, 3, 4}, donc c’est une partie (sous-ensemble).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":22364,"question":"Soit E = {x, y, z}. Que vaut P(E), l’ensemble des parties de E ?","option_a":"{{x}, {y}, {z}}","option_b":"{{x,y}, {x,z}, {y,z}, {x,y,z}}","option_c":"{{}, {x}, {y}, {z}, {x,y}, {x,z}, {y,z}, {x,y,z}}","option_d":"{{x,y,z}}","option_e":"","option_f":"","bonne_reponse":"c","explication":"P(E) inclut l’ensemble vide, les singletons, les paires, et l’ensemble E lui-même.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"{{x}, {y}, {z}}\", \"b\": \"{{x,y}, {x,z}, {y,z}, {x,y,z}}\", \"c\": \"{{","_debug_options_count":4},{"id":22365,"question":"Une application f : E → F est surjective si et seulement si tout élément de F a au moins un antécédent dans E.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C’est la définition d’une application surjective : chaque élément de F est l’image d’au moins un élément de E.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":22366,"question":"Soit A = {1, 2} et B = {3, 4}. Que vaut A ∪ B ?","option_a":"{1, 2, 3}","option_b":"{1, 2, 3, 4}","option_c":"{1, 3, 2, 4}","option_d":"∅","option_e":"","option_f":"","bonne_reponse":"b","explication":"L’union A ∪ B est l’ensemble de tous les éléments appartenant à A ou à B, ici {1, 2, 3, 4}.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"{1, 2, 3}\", \"b\": \"{1, 2, 3, 4}\", \"c\": \"{1, 3, 2, 4}\", \"d\": \"∅\"}","_debug_options_count":4},{"id":22367,"question":"Soit f : {1, 2} → {a, b} définie par f(1)=a et f(2)=a. Cette application est-elle injective ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"f n’est pas injective car deux éléments distincts de l’ensemble de départ (1 et 2) ont la même image (a).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":22368,"question":"Combien d’éléments possède l’ensemble des parties d’un ensemble à 4 éléments ?","option_a":"4","option_b":"8","option_c":"16","option_d":"32","option_e":"","option_f":"","bonne_reponse":"c","explication":"Un ensemble à n éléments possède 2ⁿ parties. Ici, n=4, donc 2⁴ = 16 parties.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"4\", \"b\": \"8\", \"c\": \"16\", \"d\": \"32\"}}","_debug_options_count":4}]