// Numbas version: exam_results_page_options
{"name": "Basic Set Theory: fun and games with the power set", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"css": "", "js": ""}, "rulesets": {}, "variable_groups": [], "extensions": [], "metadata": {"description": "<p>Slightly harder introductory exercises about the power set.</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "ungrouped_variables": ["n"], "variables": {"n": {"name": "n", "description": "", "group": "Ungrouped variables", "definition": "random(0..2)", "templateType": "anything"}}, "parts": [{"failureRate": 1, "customMarkingAlgorithm": "", "unitTests": [], "type": "jme", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "stepsPenalty": 0, "prompt": "<p>$P(A)$ contains at least the elements $\\left\\{\\right\\}$ and $A$. The case where $\\left\\{\\right\\} = A$ is particularly interesting. What is $P(\\left\\{\\right\\})$?</p>", "checkVariableNames": false, "steps": [{"failureRate": 1, "customMarkingAlgorithm": "", "unitTests": [], "type": "jme", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "prompt": "<p>What is $\\left|\\left\\{\\right\\}\\right|$?</p>", "checkVariableNames": false, "customName": "", "showPreview": true, "useCustomName": false, "valuegenerators": [], "showFeedbackIcon": true, "vsetRangePoints": 5, "variableReplacements": [], "checkingType": "absdiff", "marks": 1, "scripts": {}, "showCorrectAnswer": true, "vsetRange": [0, 1], "answer": "0"}, {"failureRate": 1, "customMarkingAlgorithm": "", "unitTests": [], "type": "jme", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "prompt": "<p>Hence, what is $\\left|P(\\left\\{\\right\\})\\right|$?</p>", "checkVariableNames": false, "customName": "", "showPreview": true, "useCustomName": false, "valuegenerators": [], "showFeedbackIcon": true, "vsetRangePoints": 5, "variableReplacements": [], "checkingType": "absdiff", "marks": 1, "scripts": {}, "showCorrectAnswer": true, "vsetRange": [0, 1], "answer": "1"}, {"useCustomName": false, "customMarkingAlgorithm": "", "unitTests": [], "showFeedbackIcon": true, "type": "information", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "prompt": "<p>Hence $P(\\left\\{\\right\\})$ is a set that only contains one element. But it must at least contain the element $\\left\\{\\right\\}$. Ponder this for a moment and then answer the question below, and remember that the NUMBAS syntax for $\\left\\{\\right\\}$ is <code>set().</code></p>", "scripts": {}, "showCorrectAnswer": true, "customName": ""}], "customName": "", "showPreview": true, "useCustomName": false, "valuegenerators": [], "showFeedbackIcon": true, "vsetRangePoints": 5, "variableReplacements": [], "checkingType": "absdiff", "marks": 1, "scripts": {}, "showCorrectAnswer": true, "vsetRange": [0, 1], "answer": "{set(set())}"}, {"failureRate": 1, "customMarkingAlgorithm": "", "unitTests": [], "type": "jme", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "stepsPenalty": 0, "prompt": "<p>If $\\left|S\\right|=\\var{n}$ then what is $\\left|P(P(P(S)))\\right|$?</p>", "checkVariableNames": false, "steps": [{"failureRate": 1, "customMarkingAlgorithm": "", "unitTests": [], "type": "jme", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "prompt": "<p>Using the handy fact, $\\left|P(S)\\right| = 2^{\\var{n}}$.</p>\n<p>Using this handy fact again, we deduce that $\\left|P(P(S))\\right| = 2^{\\left|P(S)\\right|}$, which is</p>", "checkVariableNames": false, "customName": "", "showPreview": true, "useCustomName": false, "valuegenerators": [], "showFeedbackIcon": true, "vsetRangePoints": 5, "variableReplacements": [], "checkingType": "absdiff", "marks": 1, "scripts": {}, "showCorrectAnswer": true, "vsetRange": [0, 1], "answer": "2^2^{n}"}, {"useCustomName": false, "customMarkingAlgorithm": "", "unitTests": [], "showFeedbackIcon": true, "type": "information", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "prompt": "<p>Use the handy fact yet again to determine $\\left|P(P(P(S)))\\right|$.</p>", "scripts": {}, "showCorrectAnswer": true, "customName": ""}], "customName": "", "showPreview": true, "useCustomName": false, "valuegenerators": [], "showFeedbackIcon": true, "vsetRangePoints": 5, "variableReplacements": [], "checkingType": "absdiff", "marks": 1, "scripts": {}, "showCorrectAnswer": true, "vsetRange": [0, 1], "answer": "2^2^2^{n}"}], "functions": {}, "statement": "<p>We use the notation $\\left|A\\right| = n$ to mean that $A$ contains $n$ elements. This is often called the <strong>cardinality</strong> of the set. Here is a handy fact about the number of elements in a power set.</p>\n<p style=\"text-align: center;\">If $\\left|A\\right| = n$ then $\\left|P(A)\\right| =2^n$.</p>\n<p>Using this fact, answer the following questions regarding the power set.</p>", "name": "Basic Set Theory: fun and games with the power set", "tags": [], "advice": "<p>Since $P(\\left\\{\\right\\}) = 2^{\\left|\\left\\{\\right\\}\\right|} = 2^0 = 1$ we know that $P(\\left\\{\\right\\})$ is a one-element set which contains $\\left\\{\\right\\}$. There is only one possible answer</p>\n<p style=\"text-align: center;\">$ \\left\\{\\left\\{\\right\\}\\right\\}$.</p>\n<hr/>\n<p>We construct the answer gradually. Since $\\left|S\\right| = \\var{n}$</p>\n<ul>\n<li>$\\left|P(S)\\right| = 2^\\var{n}$</li>\n<li>$\\left|P(P(S))\\right| = 2^{\\left|P(S)\\right|} = 2^{2^\\var{n}}$</li>\n<li>$\\left|P(P(P(S)))\\right| = 2^{\\left|P(P(S))\\right|} = 2^{2^{2^\\var{n}}}$.</li>\n</ul>\n<p></p>\n<p></p>", "type": "question", "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}, {"name": "Sean Gardiner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2443/"}]}]}], "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}, {"name": "Sean Gardiner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2443/"}]}