// Numbas version: exam_results_page_options {"name": "Power set - empty set and |P(P(P(S))|", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Power set - empty set and |P(P(P(S))|", "tags": [], "metadata": {"description": "

Slightly harder introductory exercises about the power set.

", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "

$\\mathcal{P}(A)$ represents the power set of A.

\n

Note that you input sets in the form set(a,b,c,d) and the empty set is input as set().

a) Using the  fact $P(\\left\\{\\right\\}) = 2^{\\left|\\left\\{\\right\\}\\right|} = 2^0 = 1$. So $P(\\left\\{\\right\\})$ is a one-element set which contains $\\left\\{\\right\\}$. There is only one possible answer
$\\left\\{\\left\\{\\right\\}\\right\\}$.
b) Rcall that if $|A|$=n$, then$|\\mathcal{P}(A)|=2^n$. \n \n c) We construct the answer gradually. Since$\\left|S\\right| = \\var{n}$\n \n •$\\left|\\mathcal{P}(S)\\right| = 2^\\var{n}= \\var{powern}$• \n •$\\left|\\mathcal{P}(\\mathcal{P}(S))\\right| = 2^{\\left|\\mathcal{P}(S)\\right|} = 2^{\\var{powern}}=\\var{powerpn}$• \n •$\\left|\\mathcal{P}(\\mathcal{P}(\\mathcal{P}(S))\\right| = 2^{\\left|\\mathcal{P}(\\mathcal{P}(S))\\right|} = 2^{\\var{powerpn}}=\\var{powerppn}$. • \n \n \n ", "rulesets": {}, "extensions": [], "variables": {"powern": {"name": "powern", "group": "Ungrouped variables", "definition": "2^n", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(0..2)", "description": "", "templateType": "anything"}, "powerpn": {"name": "powerpn", "group": "Ungrouped variables", "definition": "2^powern", "description": "", "templateType": "anything"}, "powerppn": {"name": "powerppn", "group": "Ungrouped variables", "definition": "2^powerpn", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["n", "powern", "powerpn", "powerppn"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "$\\mathcal{P}(A)$contains at least the elements$\\left\\{\\right\\}$and$A$. The case where$\\left\\{\\right\\} = A$is particularly interesting. What is$\\mathcal{P}(\\left\\{\\right\\})$? ", "answer": "{set(set())}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": []}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": " What is$|\\mathcal{P}(\\{\\})|$? ", "minValue": "1", "maxValue": "1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": " If$\\left|S\\right|=\\var{n}$then what is$\\left|\\mathcal{P}(\\mathcal{P}(\\mathcal{P}(S)))\\right|\$?