Testing the understanding of the formal definition of $A\\subseteq B$.
", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "In this question we'll be practicing the notion of a subset.
", "advice": "The feedback on the harder parts c) to f) gives detailed explanations. Accessible after \"Submit part\" and then \"Show feedback\".
", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "First, we recall the definition of a subset. Let $A$ and $B$ be two sets. We say that $A$ is a subset of $B$ and write $A\\subseteq B$ if ...
(check all that apply; wrong answers will give negative marks)
... for all $x$ we have that $x$ is an element of $A$ and $x$ is an element of $B$.
", "... $\\forall x\\in A~:~A\\subseteq B$.
", "... $\\forall x~:~x\\in A~\\Rightarrow~x\\in B$.
", "... $x$ is an element of $B$ whenever $x$ is an element of $A$.
", "... $x\\in A~\\wedge~x\\in B$.
"], "matrix": ["-3", "-3", "3", "3", "-3"], "distractors": ["", "", "", "", ""]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Let $A=\\{-1,-3,-9\\}$ and $B=\\{m\\mid m \\text{ is an odd integer }\\}$.
\nThen $A\\subseteq B$ is
", "minMarks": 0, "maxMarks": "4", "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["true
", "false
"], "matrix": ["4", 0], "distractors": ["", ""]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Let $A=\\{0,1,3,7,1023\\}$ and $B=\\{m\\mid m =2^n-1\\text{ for some }n\\in\\mathbb N\\,\\}$.
\nThen $A\\subseteq B$ is
", "minMarks": 0, "maxMarks": "4", "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["true
", "false
"], "matrix": ["0", "4"], "distractors": ["Notice that $0$ is not an element of $B$ because, although $0=2^0-1$, it is not the case that $0\\in\\mathbb N$.", "Indeed $A\\not\\subseteq B$. Note that $0$ is not an element of $B$ because, although $0=2^0-1$, it is not the case that $0\\in\\mathbb N$."]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Let $A=\\emptyset$ (the empty set) and $B=\\{0\\}$.
\nThen $A\\subseteq B$ is
", "minMarks": 0, "maxMarks": "3", "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["true
", "false
"], "matrix": ["3", "0"], "distractors": ["Indeed, the empty set is a subset of any set.", "The empty set is a subset of any set (check the definition of subset)."]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Let $A=\\{n\\in\\mathbb N\\mid n \\text{ is even }\\}$ and $B=\\{n\\in\\mathbb N\\mid n^2 \\text{ is even }\\}$.
\nThen $A\\subseteq B$ is
", "minMarks": 0, "maxMarks": "4", "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["true
", "false
"], "matrix": ["4", "0"], "distractors": ["To see that $A$ is indeed a subset of $B$ note that any element of $A$ is of the form $2k$ for some natural number $k$. Since $(2k)^2=2(2k^2)$ the element is also in $B$. ", "To see that $A$ is indeed a subset of $B$ note that any element of $A$ is of the form $2k$ for some natural number $k$. Since $(2k)^2=2(2k^2)$ the element is also in $B$. "]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Let $A=\\{0\\}$ and $B=\\{\\mathbb N\\}$.
\nThen $A\\subseteq B$ is
", "minMarks": 0, "maxMarks": "4", "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["true
", "false
"], "matrix": ["0", "4"], "distractors": ["Both $A$ and $B$ are singletons (that is, contain only a single element). The two elements they contain are different: $A$ contains the number $0$, whereas $B$ contains the set $\\mathbb N$ of all natural numbers. Therefore $A\\not\\subseteq B$.", "Both $A$ and $B$ are singletons (that is, contain only a single element). The two elements they contain are different: $A$ contains the number $0$, whereas $B$ contains the set $\\mathbb N$ of all natural numbers. Therefore $A\\not\\subseteq B$."]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Bernhard von Stengel", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1054/"}, {"name": "Peter Allen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2724/"}]}]}], "contributors": [{"name": "Bernhard von Stengel", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1054/"}, {"name": "Peter Allen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2724/"}]}