// Numbas version: finer_feedback_settings
{"name": "Least upper bound and greatest lower bound of sets", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s1", "description": ""}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "-random(1..9)", "name": "a1", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "a", "description": ""}, "c6": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..6)", "name": "c6", "description": ""}, "b4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "r5^2*a4", "name": "b4", "description": ""}, "lub3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(s1>0,a2+b2,a2)", "name": "lub3", "description": ""}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s2", "description": ""}, "perhaps2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(s1>0,'Yes','No')", "name": "perhaps2", "description": ""}, "a7": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(10..20)", "name": "a7", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "b", "description": ""}, "a8": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(7..20)", "name": "a8", "description": ""}, "a5": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "name": "a5", "description": ""}, "b6": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5)", "name": "b6", "description": ""}, "a6": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..5)", "name": "a6", "description": ""}, "perhaps1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(s1<0,'Yes','No')", "name": "perhaps1", "description": ""}, "sg": {"group": "Ungrouped variables", "templateType": "anything", "definition": "0", "name": "sg", "description": ""}, "a2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(1..9)", "name": "a2", "description": ""}, "s9": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(b6<0,1,-1)", "name": "s9", "description": ""}, "something": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(s1=-1,'positive','negative')", "name": "something", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "name": "c", "description": ""}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s*random(1..9)", "name": "d", "description": ""}, "m": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..5)", "name": "m", "description": ""}, "b7": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "name": "b7", "description": ""}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "name": "b1", "description": ""}, "b2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "name": "b2", "description": ""}, "b5": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(1..9)", "name": "b5", "description": ""}, "ans4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "2*a4*r5", "name": "ans4", "description": ""}, "mo": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(s1=1,'de','in')", "name": "mo", "description": ""}, "r": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..6)", "name": "r", "description": ""}, "a4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "a4", "description": ""}, "s": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s", "description": ""}, "b8": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a8-random(1..5)", "name": "b8", "description": ""}, "r5": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2,3,4,5)", "name": "r5", "description": ""}, "glb3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(s1>0,a2,a2-b2)", "name": "glb3", "description": ""}, "glb7": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround((sqrt(1+a7)-sqrt(1+b7)),2)", "name": "glb7", "description": ""}, "glb6": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(a6*b6^2-2*a6*b6^2+4*a6^2*c6)/(4*a6^2)", "name": "glb6", "description": ""}}, "ungrouped_variables": ["ans4", "b4", "b5", "b6", "b7", "b1", "b2", "b8", "s9", "s2", "s1", "glb7", "glb6", "glb3", "a1", "a2", "a5", "a4", "a7", "a6", "a8", "c6", "a", "r5", "c", "b", "lub3", "d", "mo", "m", "s", "r", "perhaps1", "perhaps2", "sg", "something"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "name": "Least upper bound and greatest lower bound of sets", "showQuestionGroupNames": false, "functions": {}, "parts": [{"showCorrectAnswer": true, "scripts": {}, "gaps": [{"answer": "{a1}/{b1}", "vsetrange": [0, 1], "scripts": {}, "answersimplification": "std", "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "<p>Enter as a fraction or integer, not as a decimal.</p>", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": "0.5", "vsetrangepoints": 5}, {"answer": "{a}/{b}", "vsetrange": [0, 1], "scripts": {}, "answersimplification": "std", "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "<p>Enter as a fraction or integer, not as a decimal.</p>", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": "0.5", "vsetrangepoints": 5}, {"displayType": "radiogroup", "choices": ["<p>Yes</p>", "<p>No</p>"], "showCorrectAnswer": true, "matrix": ["0.5", 0], "distractors": ["", ""], "variableReplacements": [], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 2, "marks": 0}, {"displayType": "radiogroup", "choices": ["<p>Yes</p>", "<p>No</p>"], "showCorrectAnswer": true, "matrix": [0, "0.5"], "distractors": ["", ""], "variableReplacements": [], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 0, "marks": 0}], "type": "gapfill", "prompt": "<p>\\[S = \\left\\{\\simplify[std]{({a}n^2+{a1})/({b}n^2+{b1})}\\;\\;|\\;\\;n \\in \\mathbb{Z} \\right\\}\\]</p>\n<p>Greatest lower bound = [[0]] (Enter as a fraction or integer, not a decimal.)</p>\n<p>Least upper bound = [[1]] (Enter as a fraction or integer, not a decimal.)</p>\n<p>Does the glb lie in the set? [[2]]</p>\n<p>Does the lub lie in the set? [[3]]</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"answer": "-infinity", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": "0.5", "vsetrangepoints": 5}, {"displayType": "radiogroup", "choices": ["<p>Yes</p>", "<p>No</p>"], "showCorrectAnswer": true, "matrix": [0, "0.5"], "distractors": ["", ""], "variableReplacements": [], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 2, "marks": 0}, {"answer": "{d}/{c}", "vsetrange": [0, 1], "scripts": {}, "answersimplification": "std", "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "<p>Enter as a fraction or integer, not as a decimal.</p>", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": "0.5", "vsetrangepoints": 5}, {"displayType": "radiogroup", "choices": ["<p>Yes</p>", "<p>No</p>"], "showCorrectAnswer": true, "matrix": [0, "0.5"], "distractors": ["", ""], "variableReplacements": [], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 2, "marks": 0}], "type": "gapfill", "prompt": "<p>\\[S = \\left\\{ x \\in \\mathbb{R}\\;|\\;\\simplify[std]{{c}x^{2m+1} &lt; {d}x^{2m}} \\right\\}\\]</p>\n<p>Greatest lower bound = [[0]]</p>\n<p>Does this lie in the set? [[1]]</p>\n<p>Least upper bound = [[2]] &nbsp;(Enter as a fraction or integer, not a decimal.)</p>\n<p>Does this lie in the set? [[3]]</p>", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"answer": "{glb3}", "vsetrange": [0, 1], "scripts": {}, "answersimplification": "std", "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "<p>Enter as a fraction or integer, not as a decimal.</p>", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": "0.5", "vsetrangepoints": 5}, {"displayType": "radiogroup", "choices": ["<p>{perhaps1}</p>", "<p>{perhaps2}</p>"], "showCorrectAnswer": true, "matrix": ["0.5", 0], "distractors": ["", ""], "variableReplacements": [], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 2, "marks": 0}, {"answer": "{lub3}", "vsetrange": [0, 1], "scripts": {}, "answersimplification": "std", "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "<p>Enter as a fraction or integer, not as a decimal.</p>", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": "0.5", "vsetrangepoints": 5}, {"displayType": "radiogroup", "choices": ["<p>Yes</p>", "<p>No</p>"], "showCorrectAnswer": true, "matrix": "if(s1>0,[0.5,0],[0,0.5])", "variableReplacements": [], "shuffleChoices": false, "scripts": {}, "maxMarks": 0, "type": "1_n_2", "minMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 2, "marks": 0}], "type": "gapfill", "prompt": "<p>\\[S = \\left\\{\\simplify[std]{{a2}+{s1}*{b2}/n^{r}}\\;\\;|\\;\\;n \\in \\mathbb{N} \\right\\}\\]</p>\n<p>Greatest lower bound = [[0]] (Enter as a fraction or integer, not a decimal.)</p>\n<p>Does this lie in the set? [[1]]</p>\n<p>Least upper bound = [[2]] (Enter as a fraction or integer, not a decimal.)</p>\n<p>Does this lie in the set? [[3]]</p>", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"answer": "{ans4}", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": "0.5", "vsetrangepoints": 5}, {"displayType": "radiogroup", "choices": ["<p>Yes</p>", "<p>No</p>"], "showCorrectAnswer": true, "matrix": ["0.5", 0], "distractors": ["", ""], "variableReplacements": [], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 2, "marks": 0}, {"answer": "infinity", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": "0.5", "vsetrangepoints": 5}, {"displayType": "radiogroup", "choices": ["<p>Yes</p>", "<p>No</p>"], "showCorrectAnswer": true, "matrix": [0, "0.5"], "distractors": ["", ""], "variableReplacements": [], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 2, "marks": 0}], "type": "gapfill", "prompt": "<p>\\[S = \\left\\{\\simplify[std]{{a4}x+{b4}/x}\\;\\;|\\;\\;x \\in \\mathbb{R},\\;\\;x \\gt 0 \\right\\}\\]</p>\n<p>Greatest lower bound = [[0]]</p>\n<p>Does this lie in the set? [[1]]</p>\n<p>Least upper bound = [[2]] $\\;\\;\\;\\;$</p>\n<p>Does this lie in the set? [[3]]</p>", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"answer": "-infinity", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": "0.5", "vsetrangepoints": 5}, {"displayType": "radiogroup", "choices": ["<p>Yes</p>", "<p>No</p>"], "showCorrectAnswer": true, "matrix": [0, "0.5"], "distractors": ["", ""], "variableReplacements": [], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 2, "marks": 0}, {"answer": "infinity", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": "0.5", "vsetrangepoints": 5}, {"displayType": "radiogroup", "choices": ["<p>Yes</p>", "<p>No</p>"], "showCorrectAnswer": true, "matrix": [0, "0.5"], "distractors": ["", ""], "variableReplacements": [], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 2, "marks": 0}], "type": "gapfill", "prompt": "<p>\\[S = \\left\\{\\simplify[std]{{a5}x^2+{b5}/x^3}\\;\\;|\\;\\;x \\in \\mathbb{R},\\;\\;x \\neq 0 \\right\\}\\]</p>\n<p>Greatest lower bound = [[0]]</p>\n<p>Does this lie in the set? [[1]]</p>\n<p>Least upper bound = [[2]] $\\;\\;\\;\\;$</p>\n<p>Does this lie in the set? [[3]]</p>", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"answer": "{glb6}", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": "0.5", "vsetrangepoints": 5}, {"displayType": "radiogroup", "choices": ["<p>Yes</p>", "<p>No</p>"], "showCorrectAnswer": true, "matrix": ["0.5", 0], "distractors": ["", ""], "variableReplacements": [], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 2, "marks": 0}, {"answer": "infinity", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": "0.5", "vsetrangepoints": 5}, {"displayType": "radiogroup", "choices": ["<p>Yes</p>", "<p>No</p>"], "showCorrectAnswer": true, "matrix": [0, "0.5"], "distractors": ["", ""], "variableReplacements": [], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 2, "marks": 0}], "type": "gapfill", "prompt": "<p>\\[S = \\left\\{\\simplify[std]{{a6}x^2+{b6}x+{c6}}\\;\\;|\\;\\;x \\in \\mathbb{R}\\right\\}\\]</p>\n<p>Greatest lower bound = [[0]] (enter as a fraction or integer, not a decimal)</p>\n<p>Does this lie in the set? [[1]]</p>\n<p>Least upper bound = [[2]] $\\;\\;\\;\\;$</p>\n<p>Does this lie in the set? [[3]]</p>", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "minValue": "{glb7-0.01}", "showCorrectAnswer": true, "variableReplacements": [], "marks": "0.5", "maxValue": "{glb7+0.01}"}, {"displayType": "radiogroup", "choices": ["<p>Yes</p>", "<p>No</p>"], "showCorrectAnswer": true, "matrix": ["0.5", 0], "distractors": ["", ""], "variableReplacements": [], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 2, "marks": 0}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "minValue": "{(a7-b7)/2}", "showCorrectAnswer": true, "variableReplacements": [], "marks": "0.5", "maxValue": "{(a7-b7)/2}"}, {"displayType": "radiogroup", "choices": ["<p>Yes</p>", "<p>No</p>"], "showCorrectAnswer": true, "matrix": [0, "0.5"], "distractors": ["", ""], "variableReplacements": [], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 2, "marks": 0}], "type": "gapfill", "prompt": "<p>\\[S = \\left\\{\\sqrt{\\simplify[std]{n^2+{a7}n}}-\\sqrt{\\simplify[std]{n^2+{b7}n}}\\;\\;|\\;\\;n \\in \\mathbb{N}\\right\\}\\]</p>\n<p>Greatest lower bound = [[0]] (to 2 decimal places)</p>\n<p>Does this lie in the set? [[1]]</p>\n<p>Least upper bound = [[2]] (to one decimal place.)</p>\n<p>Does this lie in the set? [[3]]</p>", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "minValue": "{a8}", "showCorrectAnswer": true, "variableReplacements": [], "marks": "0.5", "maxValue": "{a8}"}, {"displayType": "radiogroup", "choices": ["<p>Yes</p>", "<p>No</p>"], "showCorrectAnswer": true, "matrix": [0, "0.5"], "distractors": ["", ""], "variableReplacements": [], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 2, "marks": 0}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "minValue": "{a8+b8}", "showCorrectAnswer": true, "variableReplacements": [], "marks": "0.5", "maxValue": "{a8+b8}"}, {"displayType": "radiogroup", "choices": ["<p>Yes</p>", "<p>No</p>"], "showCorrectAnswer": true, "matrix": ["0.5", 0], "distractors": ["", ""], "variableReplacements": [], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 2, "marks": 0}], "type": "gapfill", "prompt": "<p>\\[S = \\left\\{\\left(\\var{a8}^n+\\var{b8}^n\\right)^{1/n}\\;\\;|\\;\\;n \\in \\mathbb{N}\\right\\}\\]</p>\n<p>Greatest lower bound = [[0]]&nbsp;</p>\n<p>Does this lie in the set? [[1]]</p>\n<p>Least upper bound = [[2]]&nbsp;</p>\n<p>Does this lie in the set? [[3]]</p>", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>For each of the following sets $S$ , state the least upper bound (lub) and the greatest lower bound (glb), where appropriate.</p>\n<p>Enter the lub as infinity i.e. type in the word infinity, if the set is not bounded above.</p>\n<p>Enter the glb as -infinity i.e. type in the word -infinity, if the set is not bounded below.</p>\n<p>$\\mathbb{N}$ denotes the set of natural numbers, $\\mathbb{Z}$ the set of integers and $\\mathbb{R}$ the set of real numbers.</p>\n<p>Also state if the lub or glb belong to the set.</p>\n<p>There are $8$ parts to this question, so you may need to scroll down to answer all parts.</p>", "tags": ["bounded above", "bounded below", "bounded set", "bounds", "checked2015", "glb", "greatest lower bound", "least upper bound", "limit", "limits", "lower bound", "lub", "MAS1701", "MAs1701", "max value", "maximum value", "min value", "minimum value", "not bounded", "sets", "upper bound"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>23/11/2015:</strong></p>\n<p>Adjusted marks available from 32 -&gt; 16</p>\n<p><strong></strong></p>\n<p><strong>4/07/2012:</strong></p>\n<p>Added tags. Corrected tags.</p>\n<p>Corrected mistake in answer to first part (minus sign missing).</p>\n<p><strong>5/07/2012:</strong></p>\n<p>There is an issue with the MCQs - this has been reported on Github.</p>\n<p>Also an issue with recognising infinity as an answer - also reported on Github.</p>\n<p>Changed to Match Text Pattern, but Correct Answer not properly displayed for $\\pm \\infty$</p>\n<p>Also an issue with reordering gaps in a gapfill - wishlist item on Github</p>\n<p>Advice display tidied up.</p>\n<p><strong>21/07/2012:</strong></p>\n<p>Error in part c first MCQ. Corrected.</p>\n<p>Instructions about using fractions and integers included.</p>\n<p>Added description.</p>\n<p>Have used Matching Expressions question typefor identifying $\\pm \\infty$ as answers.</p>\n<p><strong>27/7/2012:</strong></p>\n<p>Added tags.</p>\n<p>Edited grammar in Advice section.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Eight questions on finding least upper bounds and greatest lower bounds of various sets.</p>"}, "advice": "<p><strong>a)</strong><br/>\\[\\begin{eqnarray*} \\simplify[std]{({a}n^2+{a1})/({b}n^2+{b1})}&amp;=&amp; \\simplify[std]{(({a} / {b}) * ({b} * n ^ 2 + {b1}) + {a1} -({a * b1} / {b})) / ({b} * n ^ 2 + {b1})}\\\\ &amp;=&amp; \\simplify[std]{{a} / {b} -({( -a1) * b + a * b1} / ({b} * ({b} * n ^ 2 + {b1})))}\\\\ \\end{eqnarray*} \\]<br/>Note that 1) the values for positive and negative values of $n$ are the same and 2) as $n$ increases this expression increases.</p>\n  <p>The greatest lower bound occurs when $n=0$ and the value is $\\displaystyle \\simplify[std]{{a1}/{b1}}$.</p>\n  <p>As $n$ increases, the value of the expression approaches as close as we like to $\\displaystyle \\simplify[std]{{a}/{b}}$ , but is always less than $\\displaystyle&nbsp;\\simplify[std]{{a}/{b}}$.</p>\n  <p>Hence the least upper bound is $\\simplify[std]{{a}/{b}}$.</p>\n  <p><strong>b)</strong><br/>\\[\\begin{eqnarray*} \\simplify[std]{{c} * x ^ {2 * m + 1}}&amp;\\lt&amp;\\simplify[std]{ {d} * x ^ {2 * m}} \\Leftrightarrow\\\\ \\simplify[std]{x ^ {2 * m} * ({c} * x -{d})} &amp;\\lt&amp; 0 \\Leftrightarrow\\\\ \\simplify[std]{{c}x-{d}} &amp;\\lt&amp; 0 \\textrm{ as }x^{\\var{2*m}} \\geq 0 \\end{eqnarray*} \\]</p>\n  <p>Hence this set is the same as the set<br/>\\[\\left \\{x \\in \\mathbb{R}\\;\\;:\\;\\;x \\lt \\simplify[std]{{d}/{c}}\\right\\}\\]<br/>This set does not have a greatest lower bound so you enter -infinity.</p>\n  <p>It does have a least upper bound and this is $\\simplify[std]{{d}/{c}}$</p>\n  <p><strong>c)</strong><br/>\\[S = \\left\\{\\simplify[std]{{a2}+{s1}*{b2}/n^{r}}\\;\\;:\\;\\;n \\in \\mathbb{N} \\right\\}\\]<br/>Let $\\displaystyle&nbsp;a_n=\\simplify[std]{{a2}+{s1}*{b2}/n^{r}}$</p>\n  <p>As $n$ increases we see that $a_n$ {mo}creases and converges to the limit $\\var{a2}$.</p>\n  <p>Hence greatest lower bound = $\\var{glb3}$ and least upper bound = $\\var{lub3}$</p>\n  <p><strong>d)</strong><br/>\\[S = \\left\\{\\simplify[std]{{a4}x+{b4}/x}\\;\\;:\\;\\;x \\in \\mathbb{R},\\;\\;x \\gt 0 \\right\\}\\]</p>\n  <p>It is clear that this set does not have a least upper bound, so we enter infinity for this value.</p>\n  <p>However it does have a lower bound as we have $\\displaystyle \\var{a4}x+\\frac{\\var{b4}}{x} \\gt 0,\\;\\;\\forall x \\gt 0 $.</p>\n  <p>To find the greatest lower bound we find the minimum value of $\\displaystyle g(x)=\\var{a4}x+\\frac{\\var{b4}}{x},\\;\\;x \\gt 0 $.</p>\n  <p>Now $\\displaystyle g'(x)=\\var{a4}-\\frac{\\var{b4}}{x^2}$ and $g'(x)=0$ when $\\displaystyle x=\\sqrt{\\frac{\\var{b4}}{\\var{a4}}} = \\var{r5}$.</p>\n  <p>(We take the positive square root as $x \\gt 0$).</p>\n  <p>It is not hard to see that this gives a minimum value for $g(x)$ and $g(\\var{r5})=\\var{ans4}$.</p>\n  <p>Hence the greatest lower bound is $\\var{ans4}$ as $g(x) \\geq \\var{ans4},\\;\\;\\forall x \\gt 0$.</p>\n  <p><strong>e)</strong></p>\n  <p>\\[S = \\left\\{\\simplify[std]{{a5}x^2+{b5}/x^3}\\;\\;:\\;\\;x \\in \\mathbb{R},\\;\\;x \\neq 0 \\right\\}\\]</p>\n  <p>This set does not have an upper bound as $\\var{a5}x^2 \\longrightarrow \\infty\\textrm{ as }x\\longrightarrow \\infty$.</p>\n  <p>Also it does not have a lower bound as if $x\\longrightarrow \\var{sg}$ through {something} values of $x$ then $ \\displaystyle\\simplify[std]{{b5}/x^3}\\longrightarrow -\\infty$.</p>\n  <p><strong>f)</strong><br/>\\[S = \\left\\{\\simplify[std]{{a6}x^2+{b6}x+{c6}}\\;\\;:\\;\\;x \\in \\mathbb{R}\\right\\}\\]</p>\n  <p>Since this is a quadratic with positive coefficient of the $x^2$ term it has a minimum value at $\\displaystyle&nbsp;x=\\simplify[std]{{-b6}/{2*a6}}$.</p>\n  <p>It follows that the minimum value and hence the glb is $\\var{glb6}$ on substituting into the quadratic.</p>\n  <p>As it is a quadratic with positive coefficient of the $x^2$ term it tends to $\\infty$ as $x \\longrightarrow \\infty$ or $-\\infty$.</p>\n  <p><strong>g)</strong><br/>\\[S = \\left\\{\\sqrt{\\simplify[std]{n^2+{a7}n}}-\\sqrt{\\simplify[std]{n^2+{b7}n}}\\;\\;:\\;\\;n \\in \\mathbb{N}\\right\\}\\]</p>\n  <p>We have:<br/>\\[\\begin{eqnarray*} \\sqrt{\\simplify[std]{n^2+{a7}n}}-\\sqrt{\\simplify[std]{n^2+{b7}n}}&amp;=&amp;\\simplify[std]{({a7 -b7} * n) / (sqrt(n ^ 2 + {a7} * n) + sqrt(n ^ 2 + {b7} * n))}\\\\ &amp;=&amp;\\simplify[std]{{a7 -b7} / (sqrt(1 + {a7} / n) + sqrt(1 + {b7} / n))}\\\\ &amp;\\lt&amp;\\simplify[std]{{a7 -b7} / 2} \\end{eqnarray*} \\]<br/>From the above, we see that $\\sqrt{\\simplify[std]{n^2+{a7}n}}-\\sqrt{\\simplify[std]{n^2+{b7}n}}$ is increasing as $n$ increases, hence the minimum value is at $n=1$ and this is the glb.</p>\n  <p>Hence glb = $\\sqrt{\\simplify[std]{1+{a7}}}-\\sqrt{\\simplify[std]{1+{b7}}}=\\var{glb7}$ to 2 decimal places.</p>\n  <p>Now as $n$ increases the terms approach $\\displaystyle&nbsp;\\simplify[std]{{a7 -b7} / 2}$, but never equal this value, hence the lub is $\\displaystyle&nbsp;\\simplify[std]{{a7 -b7} / 2}$.</p>\n  <p><strong>h)</strong><br/>Since $\\var{a8}$ and $\\var{b8}$ are positive it is true that $(\\var{a8}+\\var{b8})^n \\geq \\var{a8}^n+\\var{b8}^n$.</p>\n  <p>(Use the binomial expansion to show this.)</p>\n  <p>Hence on taking the nth roots of both sides we have $\\var{a8}+\\var{b8}=\\var{a8+b8} \\geq ( \\var{a8}^n+\\var{b8}^n)^{1/n}$.</p>\n  <p>So we see that $\\var{a8+b8}$ is an upper bound for the set and it is the lub as we get this value for $n=1$.</p>\n  <p>Now $\\displaystyle \\lim_{n \\to \\infty}\\left(\\var{a8}^n+\\var{b8}^n\\right)^{1/n}=\\var{a8}$ and as $\\var{a8} \\lt \\left(\\var{a8}^n+\\var{b8}^n\\right)^{1/n},\\;\\;\\forall n$ we see that $\\var{a8}$ is the glb, but does not belong to the set.</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/"}]}