// Numbas version: finer_feedback_settings
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Cartesian Products and Complements", "functions": {"mod_set": {"type": "list", "language": "javascript", "definition": "//returns all integers which are divisible by c betweeen a and b\nvar l=[];\nfor(var i=a;i<b+1;i++) \n{ if (i%c==0){l.push(i);}\n}\nreturn l;\n", "parameters": [["a", "number"], ["b", "number"], ["c", "number"]]}}, "showQuestionGroupNames": false, "parts": [{"prompt": "<p>$(A^c\\cap B^c) \\times (A\\cap B)=\\;$[[0]]</p>", "scripts": {}, "gaps": [{"answer": "{set3}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "<p>$(U\\times A)^c\\cap (U\\times B)^c\\cap (A\\times U)^c\\cap (B\\times U)^c=\\;$[[0]]</p>", "scripts": {}, "gaps": [{"answer": "{set4}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "<p>$(A-B)\\times (A\\cup B)^c=\\;$[[0]]</p>", "scripts": {}, "gaps": [{"answer": "{set5}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "<p>$[ (A \\cup B)\\times U]^c \\cap [ U \\times (A \\cap B) ]=\\;$[[0]]</p>", "scripts": {}, "gaps": [{"answer": "{set6}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "<p>$(A^c - B)\\times (B^c-A)=\\;$[[0]]</p>", "scripts": {}, "gaps": [{"answer": "{set7}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": "<p>In this question the universal set is $U=\\var{univ}$.</p>\n<p>Let $A=\\var{set1}$ and let&nbsp; $B=\\var{set2}$.</p>\n<p>For subsets $A$ and $B$ of $U$, the universal set for the Cartesian product $A\\times B$ is $U\\times U$.</p>\n<p>List the elements of the following sets.&nbsp;</p>\n<p>Note that you input sets in the form <code>set(a,b,c,d)</code> .</p>\n<p>For example <code> set(1,2,3)</code>gives the set $\\{1,2,3\\}$.</p>\n<p>Element $(a,b)$ of a Cartesian product is entered, and represented as $[a,b]$.</p>\n<p>For example <code> set([1,1],[1,2],[2,3])</code>gives the set $\\{[1,1], [1,2], [2,3]\\}$.</p>\n<p>The empty set is input as&nbsp;<code>set()</code>.</p>", "tags": [], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<h4>a)</h4>\n<p>$A^c$ is the set of $U-A$ of all elements in $U$ and not in $A$, so $A^c = \\var{univ - set1}$.</p>\n<p>$B^c$ is the set of $U-A$ of all elements in $U$ and not in $B$, so $B^c = \\var{univ - set1}$.</p>\n<p>$A^c \\cap B^c$ is the set of all elements present in both $A^c$&nbsp;and $B^c$. This is equivalent to the set of all elements in neither $A$ nor $B$, i.e. $\\var{(univ-set1) and (univ-set2)}$.</p>\n<p>$A \\cap B$ is the set of all elements present in both $A$ and $B$, i.e $\\var{set1 and set2}$.</p>\n<p>So $(A^c \\cap B^c) \\times (A \\cap B)$ is the set of all pairs $(x,y)$, where $x$ is in $A^c \\cap B^c$, and $y$ is in $A \\cap B$.</p>\n<h4>b)</h4>\n<p>$(U \\times A)^c$ is the set of all pairs $(x,y)$ in $U \\times U$ which are not in $U \\times A$. Since $U$ is the universal set, this is&nbsp;equivalent to $U \\times (A^c)$, the product of $U$ with the set of elements not in $A$.</p>\n<p>Similarly, $(U \\times B)^c$ is equivalent to $U \\times (B^c)$.</p>\n<p>Again because $U$ is the universal set, $(U \\times A)^c \\cap (U \\times B)^c = U \\times (A^c \\cap B^c)$.</p>\n<p>By a similar argument,&nbsp;$(A \\times U)^c \\cap (B \\times U)^c = (A^c \\cap B^c) \\times U$.</p>\n<p>So $(U\\times A)^c\\cap (U\\times B)^c\\cap (A\\times U)^c\\cap (B\\times U)^c$ is equivalent to $(A^c \\cap B^c) \\times (A^c \\cap B^c)$. That is, the set of all pairs of two elements that are in neither $A$ nor $B$.</p>\n<h4>c)</h4>\n<p>$A-B$ is the set of elements which are in $A$ but not $B$, i.e. $\\var{set1-set2}$.</p>\n<p>$(A \\cup B)^c$ is the set of elements $U - (A \\cup B)$ which are in $U$ and not in&nbsp;$A \\cup B$, so $(A \\cup B)^c = \\var{univ-(set1 or set2)}$.</p>\n<h4>d)</h4>\n<p>$[(A \\cup B) \\times U]^c$ is equivalent to $(A \\cup B)^c \\times U$.</p>\n<p>So $[(A \\cup B) \\times U]^c \\cap [U \\times (A \\cap B)] = (A \\cup B)^c \\times (A \\cap B)$.</p>\n<h4>e)</h4>\n<p>$A^c-B$ is the set of all elements which are in $A^c$ but not $B$. That's equivalent to the set of elements which are in neither $A$ nor $B$, i.e. $(A \\cup B)^c = \\var{univ-(set1 or set2)}$.</p>\n<p>Similarly, $B^c - A = (B \\cup A)^c = (A \\cup B)^c = \\var{univ-(set1 or set2)}$.</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}