Asks the student to give the images/preimages of subsets of the domain/range of a map.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Gegeben ist die folgende Abbildung $f\\colon$ $\\{1, \\dots, \\var{card_x}\\}$$\\{ 1\\}$ $\\to$ $\\{ 1, \\dots, \\var{card_y}\\}$$\\{1\\}$:
\n{vector_plot()}
\nGeben Sie Mengen in der Form set(1,2,3) (statt $\\{1,2,3\\}$) an. Die leere Menge können Sie als set() eingeben.
a), b) Um das Bild einer Teilmenge (des Definitionsbereichs) zu bestimmen, suchen Sie alle Endpunkt von Pfeilen, die in der gegebenen Menge starten.
\nc), d) Um das Urbild einer Teilmenge (des Wertebereichs) zu bestimmen, suchen Sie alle Anfangspunkt von Pfeilen, die in der gegebenen Menge enden.
", "rulesets": {}, "extensions": ["jsxgraph"], "variables": {"card_x": {"name": "card_x", "group": "Ungrouped variables", "definition": "random(3..8)", "description": "", "templateType": "anything"}, "card_y": {"name": "card_y", "group": "Ungrouped variables", "definition": "random(1..8)", "description": "", "templateType": "anything"}, "q1": {"name": "q1", "group": "Ungrouped variables", "definition": "random(1..card_y)", "description": "", "templateType": "anything"}, "q2": {"name": "q2", "group": "Ungrouped variables", "definition": "random(1..card_y)", "description": "", "templateType": "anything"}, "q3": {"name": "q3", "group": "Ungrouped variables", "definition": "random(1..card_y)", "description": "", "templateType": "anything"}, "q4": {"name": "q4", "group": "Ungrouped variables", "definition": "random(1..card_y)", "description": "", "templateType": "anything"}, "q5": {"name": "q5", "group": "Ungrouped variables", "definition": "random(1..card_y)", "description": "", "templateType": "anything"}, "q6": {"name": "q6", "group": "Ungrouped variables", "definition": "random(1..card_y)", "description": "", "templateType": "anything"}, "marking": {"name": "marking", "group": "Ungrouped variables", "definition": "[if(is_injective, 1, -1), if(is_surjective, 1, -1), if(is_injective && is_surjective ,1,-1), if(!is_injective && !is_surjective, 1, -1)]", "description": "", "templateType": "anything"}, "is_surjective": {"name": "is_surjective", "group": "Ungrouped variables", "definition": "set(take(card_x, true, x, [q1, q2, q3, q4, q5, q6]))=set(1..card_y)", "description": "", "templateType": "anything"}, "is_injective": {"name": "is_injective", "group": "Ungrouped variables", "definition": "len(set(take(card_x, true, x, [q1, q2, q3, q4, q5, q6])))=card_x", "description": "", "templateType": "anything"}, "card_x_gt_1": {"name": "card_x_gt_1", "group": "Ungrouped variables", "definition": "card_x>1", "description": "", "templateType": "anything"}, "card_y_gt_1": {"name": "card_y_gt_1", "group": "Ungrouped variables", "definition": "card_y>1", "description": "", "templateType": "anything"}, "q7": {"name": "q7", "group": "Ungrouped variables", "definition": "random(1..card_y)", "description": "", "templateType": "anything"}, "q8": {"name": "q8", "group": "Ungrouped variables", "definition": "random(1..card_y)", "description": "", "templateType": "anything"}, "F1": {"name": "F1", "group": "Part a)", "definition": "shuffle_together([1..card_x, take(card_x, true, x, [q1, q2, q3, q4, q5, q6, q7, q8])])", "description": "", "templateType": "anything"}, "card_subset": {"name": "card_subset", "group": "Part a)", "definition": "random(1..card_x)", "description": "", "templateType": "anything"}, "subset": {"name": "subset", "group": "Part a)", "definition": "set(sort(take(card_subset, true, x, F1[0])))", "description": "", "templateType": "anything"}, "image": {"name": "image", "group": "Part a)", "definition": "set(sort(take(card_subset, true, x, F1[1])))", "description": "", "templateType": "anything"}, "subsetx": {"name": "subsetx", "group": "Part b)", "definition": "set(sort(take(card_subsetx, true, x, F1x[0])))", "description": "", "templateType": "anything"}, "card_subsetx": {"name": "card_subsetx", "group": "Part b)", "definition": "random(1..card_x)", "description": "", "templateType": "anything"}, "imagex": {"name": "imagex", "group": "Part b)", "definition": "set(sort(take(card_subsetx, true, x, F1x[1])))", "description": "", "templateType": "anything"}, "F1x": {"name": "F1x", "group": "Part b)", "definition": "shuffle_together([1..card_x, values])", "description": "", "templateType": "anything"}, "subsetc": {"name": "subsetc", "group": "Part c)", "definition": "set(sort(take(random(1..card_y), true, x, shuffle(1..card_y))))", "description": "", "templateType": "anything"}, "preimagec": {"name": "preimagec", "group": "Part c)", "definition": "set(sort(filter(values[x-1] in subsetc, x, 1..card_x)))", "description": "", "templateType": "anything"}, "values": {"name": "values", "group": "Ungrouped variables", "definition": "take(card_x, true, x, [q1,q2,q3,q4,q5,q6,q7,q8])", "description": "", "templateType": "anything"}, "subsetd": {"name": "subsetd", "group": "Part d)", "definition": "set(sort(take(random(1..card_y), true, x, shuffle(1..card_y))))", "description": "", "templateType": "anything"}, "preimaged": {"name": "preimaged", "group": "Part d)", "definition": "set(sort(filter(values[x-1] in subsetd, x, 1..card_x)))", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "!(subset=subsetx) && !(subsetc=subsetd)", "maxRuns": 100}, "ungrouped_variables": ["values", "card_x", "card_y", "q1", "q2", "q3", "q4", "q5", "q6", "marking", "is_surjective", "is_injective", "card_x_gt_1", "card_y_gt_1", "q7", "q8"], "variable_groups": [{"name": "Part a)", "variables": ["F1", "card_subset", "subset", "image"]}, {"name": "Part b)", "variables": ["subsetx", "card_subsetx", "imagex", "F1x"]}, {"name": "Part c)", "variables": ["subsetc", "preimagec"]}, {"name": "Part d)", "variables": ["subsetd", "preimaged"]}], "functions": {"vector_plot": {"parameters": [], "type": "html", "language": "javascript", "definition": "var card_x = Numbas.jme.unwrapValue(question.scope.variables.card_x);\nvar card_y = Numbas.jme.unwrapValue(question.scope.variables.card_y);\nvar q1 = Numbas.jme.unwrapValue(question.scope.variables.q1);\nvar q2 = Numbas.jme.unwrapValue(question.scope.variables.q2);\nvar q3 = Numbas.jme.unwrapValue(question.scope.variables.q3);\nvar q4 = Numbas.jme.unwrapValue(question.scope.variables.q4);\nvar q5 = Numbas.jme.unwrapValue(question.scope.variables.q5);\nvar q6 = Numbas.jme.unwrapValue(question.scope.variables.q6);\nvar q7 = Numbas.jme.unwrapValue(question.scope.variables.q7);\nvar q8 = Numbas.jme.unwrapValue(question.scope.variables.q8);\n\nvar div = Numbas.extensions.jsxgraph.makeBoard(\n '700px', '350px',\n {\n boundingBox:[-2, Math.max(card_x, card_y)+2.5, 5, -2],\n grid: false,\n axis:false\n });\n \nvar board = div.board;\n\nboard.create('ellipse', [[0, 0.5], [0, card_x+0.5], [0.3, card_x/2]]);\nboard.create('ellipse', [[3, 0.5], [3, card_y+0.5], [3.3, card_y/2]]);\n\n// points\nfor(let i=1; i <= card_x; i++) {\n board.create(\n 'point', [0, i],\n {\n size: 3,\n name: '' + i,\n fixed: true,\n showInfobox: false,\n label: {fontsize:16,offset:[-20,0]}\n });\n}\n\nfor(let i=1; i <= card_y; i++) {\n board.create(\n 'point', [3, i],\n {\n size: 3,\n name: '' + i,\n fixed: true,\n showInfobox: false,\n label: {fontsize:16,offset:[10,0]}\n });\n}\n\nvalues = [q1, q2, q3, q4, q5, q6, q7, q8];\nfor(let i=1; i <= card_x; i++) {\n board.create(\n 'line',[[0, i], [3, values[i-1]]], \n {\n straightFirst: false,\n straightLast: false,\n fixed: true,\n strokeWidth:2,\n strokeColor:'blue',\n lastArrow:true,\n touchFirstPoint: true,\n touchLastPoint: true,\n layer: 10\n });\n}\n\nreturn div"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Geben Sie das Bild der Menge $\\simplify{{subset}}$ an:
\n$f(\\simplify{{subset}}) =$
", "answer": "{image}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Geben Sie das Bild der Menge $\\simplify{{subsetx}}$ an:
\n$f(\\simplify{{subsetx}}) =$
", "answer": "{imagex}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Geben Sie das Urbild der Menge $\\simplify{{subsetc}}$ an.
\n$f^{-1}(\\simplify{{subsetc}})=$
", "answer": "{preimagec}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Geben Sie das Urbild der Menge $\\simplify{{subsetd}}$ an.
\n$f^{-1}(\\simplify{{subsetd}})=$
", "answer": "{preimaged}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Ulrich G\u00f6rtz", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7603/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Ulrich G\u00f6rtz", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7603/"}]}