// 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, "typeendtoleave": 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.
\nNote 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
\n$ \\left\\{\\left\\{\\right\\}\\right\\}$.
\nb) Rcall that if $|A|$=n$, then $|\\mathcal{P}(A)|=2^n$.
\nc) We construct the answer gradually. Since $\\left|S\\right| = \\var{n}$
\n$\\mathcal{P}(A)$ 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 $|\\mathcal{P}(\\{\\})|$?
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", "answer": "2^2^2^{n}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": []}], "contributors": [{"name": "Lois Rollings", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/326/"}]}]}], "contributors": [{"name": "Lois Rollings", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/326/"}]}