// Numbas version: exam_results_page_options
{"name": "Relations: Partial Ordering and Hasse Diagram", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [["question-resources/hasse.png", "/srv/numbas/media/question-resources/hasse.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Relations: Partial Ordering and Hasse Diagram", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Another important relation is called a <strong>partial order</strong>. We say that a relation $\\leq$ is a partial order when it has the three characteristic qualities:</p>\n<ul>\n<li>$a\\leq a$ always</li>\n<li>if $a\\leq b$ and $b \\leq a$ then $b=a$</li>\n<li>if $a \\leq b$ and $b \\leq c$ then $a \\leq c$.</li>\n</ul>\n<p>Typically, we take a set together with partial ordering such as \"less than or equal to\" $\\leq$, \"subset\" $\\subseteq$ or \"divides\" $|$. Together, the partial ordering and set are called a <strong>partially ordered set</strong> or <strong>poset</strong>.</p>\n<p>Instead of an arrow diagram, we use a Hasse Diagram to visualise a poset. This communicates the relation between elements of the set with as few edges as possible.</p>\n<p>Consider the set</p>\n<p style=\"text-align: center;\">$A = \\left\\{&nbsp;\\left\\{1\\right\\} ,\\left\\{2 \\right\\} ,\\left\\{3 \\right\\},\\left\\{1,2 \\right\\} ,\\left\\{ 3,4\\right\\},\\left\\{1,2,4 \\right\\},\\left\\{1,3,4 \\right\\},\\left\\{1,2,4,5 \\right\\},\\left\\{ 1,2,3,5 \\right\\} \\right\\} $</p>\n<p style=\"text-align: left;\">together with the partial ordering $\\subseteq$.</p>", "advice": "<p>See the steps section above for a general description for how to design Hasse diagrams.</p>\n<p>A good rule of thumb is to work from the bottom up. Join all 1-element sets to those 2-element sets that contain them as a subset. Then do the same for 2-element sets to 3-element sets, and 3-element sets to 4-element sets.</p>\n<p>Next check if any of the 1-element sets are contained in any of the 3-element sets that they are not already connected to by a path of edges. If they are, join these vertices. Do the same for 2-element sets contained in 4-element sets.</p>\n<p>Finally, check if any of the 1-element sets are contained in any of the 4-element sets that they are not already connected to by a path of edges.</p>", "rulesets": {}, "extensions": ["jsxgraph"], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "random(0.1 .. 1#0.1)", "description": "", "templateType": "randrange"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(0.1 .. 1#0.1)", "description": "", "templateType": "randrange"}}, "variablesTest": {"condition": "a<>b", "maxRuns": 100}, "ungrouped_variables": ["a", "b"], "variable_groups": [], "functions": {"createNetwork": {"parameters": [], "type": "html", "language": "javascript", "definition": "// This function creates the board and sets it up, then returns an\n// HTML div tag containing the board.\n  \n// The line is described by the equation \n//   y = a*x + b\n\n// This function takes as its parameters the coefficients a and b,\n// and the coordinates (x2,y2) of a point on the line.\n\n// First, make the JSXGraph board.\n// The function provided by the JSXGraph extension wraps the board up in \n// a div tag so that it's easier to embed in the page.\nvar div = Numbas.extensions.jsxgraph.makeBoard('300px','400px',\n{boundingBox: [0,4.5,1,0.5],\n axis: false,\n showNavigation: false,\n grid: false\n});\n\n//console.log(\"Extensions\")\n//console.log(Numbas.extensions);\n\n// div.board is the object created by JSXGraph, which you use to \n// manipulate elements\n\nvar  redColor='#ff0000';\nvar blueColor='#0000ff';\nvar grayColor='#9b9b9b';\n\nvar PointStyle={face:'o', size:4, fixed:false,label:{fontSize:20}};\nvar LineStyle={fixed:true,straightFirst:false,straightLast:false,strokeWidth:2,dash:0,strokeColor:blueColor};\n\n\nquestion.parts[0].trueEdges=new Set([\"{1,2,4,5}gt{1,2,4}\", \"{1,2,3,5}gt{1,2}\", \"{1,2,3,5}gt{3}\", \"{1,3,4}gt{3,4}\", \"{1,3,4}gt{1}\", \"{1,2,4}gt{1,2}\", \"{1,2}gt{1}\", \"{1,2}gt{2}\", \"{3,4}gt{3}\"]);\n\nquestion.parts[0].allEdges = new Set([\"{1,2,4,5}gt{1,2,4}\", \"{1,2,4,5}gt{1,2}\", \"{1,2,4,5}gt{1}\", \"{1,2,4,5}gt{2}\", \"{1,2,3,5}gt{1,2}\", \"{1,2,3,5}gt{1}\", \"{1,2,3,5}gt{2}\", \"{1,2,3,5}gt{3}\", \"{1,3,4}gt{3,4}\", \"{1,3,4}gt{1}\", \"{1,3,4}gt{3}\", \"{1,2,4}gt{1,2}\", \"{1,2,4}gt{1}\", \"{1,2,4}gt{2}\", \"{1,2}gt{1}\", \"{1,2}gt{2}\", \"{3,4}gt{3}\"]);\n\nquestion.parts[0].removableEdges = new Set([\"{1,2,4,5}gt{1,2}\", \"{1,2,4,5}gt{1}\", \"{1,2,4,5}gt{2}\", \"{1,2,4,5}gt{1}\", \"{1,2,4,5}gt{2}\", \"{1,2,3,5}gt{1}\", \"{1,2,3,5}gt{2}\", \"{1,3,4}gt{3}\", \"{1,2,4}gt{1}\", \"{1,2,4}gt{2}\"]);\n\nquestion.parts[0].constructedEdges=new Set();\n\nvar lineDict={};\n\nvar point0selected=false\nvar point0\n\n\nfunction changePointStyle(){\n    //console.log(this);\n    if(this.selected==true){\n      this.selected=false;\n      this.setAttribute({fillColor:redColor,strokeColor:redColor});\n      \n      point0selected=false\n      \n    } else {\n      this.selected=true ;\n      this.setAttribute({fillColor:blueColor,strokeColor:blueColor});\n      \n      if(point0selected==false){\n          point0selected=true\n          point0=this\n      } else {\n        if(point0.Y()>this.Y()){\n            lineName=point0.name+\"gt\"+this.name;\n        } else if (point0.Y()<this.Y()) {\n            lineName=this.name+\"gt\"+point0.name   \n        } else if (this.name<point0.name){\n            lineName=this.name+\"gt\"+point0.name \n        } else {\n            lineName=point0.name+\"gt\"+this.name \n        }\n        \n        if (question.parts[0].constructedEdges.has(lineName)){\n            \n            board.removeObject(lineDict[lineName]);\n            question.parts[0].constructedEdges.delete(lineName);\n        \n        } else {\n\n            line=board.create('line',[point0,this],LineStyle);\n            console.log(line)\n            lineDict[lineName]=line\n            line.name=lineName;\n            question.parts[0].constructedEdges.add(lineName);\n            console.log(\"LineCreated\");\n            console.log(question.parts[0].constructedEdges);\n        \n        }\n        \n        point0selected=false\n        point0.selected=false\n        \n        point0.setAttribute({fillColor:redColor,strokeColor:redColor});\n        \n        this.selected=false;\n        this.setAttribute({fillColor:redColor,strokeColor:redColor});\n      }\n      \n    }\n \n};\n\nfunction resetYposition(){\n  this.moveTo([this.X(),this.nativeY]);\n};\n\nvar board = div.board;  \n\nvar PointInfoArray=[[[ 0.33, 4.00],\"{1,2,4,5}\"]\n                   ,[[ 0.66, 4.00],\"{1,2,3,5}\"]\n                    \n                   ,[[ 0.25, 3.00],\"{1,2,4}\"  ]\n                   ,[[ 0.75, 3.00],\"{1,3,4}\"  ]\n                    \n                   ,[[ 0.60, 2.00],\"{1,2}\"    ]\n                   ,[[ 0.30, 2.00],\"{3,4}\"    ]\n                    \n                   ,[[ 0.75, 1.00],\"{1}\"      ]\n                   ,[[ 0.50, 1.00],\"{2}\"      ]\n                   ,[[ 0.25, 1.00],\"{3}\"      ]];\n\nPointArray=[];\nfor (i=0;i<PointInfoArray.length;i++){\n    // console.log(PointInfoArray[i][1]);\n    PointStyle.name=PointInfoArray[i][1];\n    PointArray[i]=board.create('point',PointInfoArray[i][0],PointStyle)\n \n    PointArray[i].selected=false\n    PointArray[i].nativeY=PointInfoArray[i][0][1]\n\n    PointArray[i].on('down',changePointStyle)\n    PointArray[i].on('drag',resetYposition)\n};\n\nreturn div;"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "extension", "useCustomName": false, "customName": "", "marks": "4", "scripts": {"mark": {"script": "var totalTrue=this.trueEdges.size;\n//console.log(this.trueEdges);\n\nvar goodSet = new Set([...this.constructedEdges].filter(x => this.trueEdges.has(x)));                       \n\nvar fractionOfTrue=Math.round(100*goodSet.size/totalTrue)/100;\nvar runningGrade=fractionOfTrue;\n\nif (fractionOfTrue==0){\n    var ratioPhrase=\"none\"\n} else if (fractionOfTrue<0.3){\n    var ratioPhrase=\"a few\"\n} else if (fractionOfTrue<0.6){\n    var ratioPhrase=\"some\"\n} else if (fractionOfTrue<0.9){\n    var ratioPhrase=\"many\"\n} else if (fractionOfTrue<1){\n    var ratioPhrase=\"almost all\"\n} else {\n    var ratioPhrase=\"all\"\n}\nthis.setCredit(runningGrade,\"You have created \"+ratioPhrase+\" of the correct edges.\");\n\nif (fractionOfTrue<1){\n    var missingSet = new Set([...this.trueEdges].filter(x => !this.constructedEdges.has(x)));  \n    // Just double checking there really is a missing element\n    if (missingSet.size>0) {\n       this.setCredit(runningGrade,\"You are missing some vital edges, for example from \"+missingSet.values().next()['value'].replace(\"gt\", \" to \")+\".\");\n    }\n}\n\nvar badSet = new Set([...this.constructedEdges].filter(x => !this.allEdges.has(x)));    \nvar badReduction=0;\n\nfor (let item of badSet) {\n    //console.log(item);\n    runningGrade=Math.max(runningGrade-0.25,0);\n    this.setCredit(runningGrade,\"You have edges that are do not represent a partial order, \"+item.replace(\"gt\", \" is connected to \")+\".\")\n};\n\nvar preredundantSet = new Set([...this.constructedEdges].filter(x => this.allEdges.has(x)));  \n//console.log(\"prered\");\n//console.log(preredundantSet);\n\nvar    redundantSet = new Set([...preredundantSet].filter(x => !this.trueEdges.has(x)));  \n\nfor (let item of redundantSet) {\n    //console.log(item);\n    runningGrade=Math.max(runningGrade-0.125,0);\n\n    this.setCredit(runningGrade,\"The edge from \"+item.replace(\"gt\", \" to \")+\" can be reduced.\");\n};\n\nthis.answered = true;", "order": "instead"}, "validate": {"script": "return true", "order": "instead"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Make a Hasse Diagram for this poset. Click on two verticies to make or destroy the edge between them.</p>\n<p>{createNetwork()}</p>", "stepsPenalty": 0, "steps": [{"type": "information", "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": "<ul>\n<li>If $A \\subsetneq B$ and there is no $C$ such that $A \\subsetneq C \\subsetneq B$, draw a line between $A$ and $B$, with $A$ positioned lower than $B$.</li>\n<li>Do not draw any lines that can be deduced by the transitive property.</li>\n<li>Do not draw any loops to indicate the reflexive property.</li>\n</ul>\n<p>For example, the poset $(\\left\\{1,2,3,4,5,6\\right\\},|)$ is a poset. Its corresponding Hasse diagram is</p>\n<p style=\"text-align: center;\"><img src=\"resources/question-resources/hasse.png\"/></p>"}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "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/"}]}