// Numbas version: finer_feedback_settings
{"name": "Quantifiers 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["all", "marks"], "name": "Quantifiers 3", "tags": ["logic", "quantifiers", "statements"], "preamble": {"css": "", "js": ""}, "advice": "<p>In the following we use the rules for negating a proposition involving predicates and quantifiers as given in the lectures.</p>\n<p><strong>a)&nbsp;</strong></p>\n<p>1. The sentence:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[0]][0]}</strong>&nbsp;</p>\n<p>can be written in predicate form with quantifiers as:</p>\n<p style=\"text-align: center;\">&nbsp;&nbsp;<strong>{all[select[0]][3]}</strong></p>\n<p><strong></strong>The negation of the sentence can be written as:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[0]][2]}</strong>&nbsp;</p>\n<p>The predicate form with quantifiers for this is:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[0]][1]}</strong></p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p>2. The sentence:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[1]][0]}</strong>&nbsp;</p>\n<p>can be written in predicate form with quantifiers as:</p>\n<p style=\"text-align: center;\">&nbsp;&nbsp;<strong>{all[select[1]][3]}</strong></p>\n<p><strong></strong>The negation of the sentence can be written as:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[1]][2]}</strong>&nbsp;</p>\n<p>The predicate form with quantifiers for this is:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[1]][1]}</strong></p>\n<p style=\"text-align: left;\"></p>\n<p>3. The sentence:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[2]][0]}</strong>&nbsp;</p>\n<p>can be written in predicate form with quantifiers as:</p>\n<p style=\"text-align: center;\">&nbsp;&nbsp;<strong>{all[select[2]][3]}</strong></p>\n<p><strong></strong>The negation of the sentence can be written as:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[2]][2]}</strong>&nbsp;</p>\n<p>The predicate form with quantifiers for this is:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[2]][1]}</strong></p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\">4. The sentence:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[3]][0]}</strong>&nbsp;</p>\n<p>can be written in predicate form with quantifiers as:</p>\n<p style=\"text-align: center;\">&nbsp;&nbsp;<strong>{all[select[3]][3]}</strong></p>\n<p><strong></strong>The negation of the sentence can be written as:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[3]][2]}</strong>&nbsp;</p>\n<p>The predicate form with quantifiers for this is:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[3]][1]}</strong></p>\n<p style=\"text-align: left;\">Similarly for Parts<strong> b)</strong> and <strong>c).</strong></p>\n<p style=\"text-align: left;\"><strong></strong></p>", "rulesets": {}, "parts": [{"stepsPenalty": "1", "maxAnswers": "4", "prompt": "<p>If you want some help in answering this question click on Show steps. You will lose a mark as one of the questions is answered for you.</p>", "matrix": "marks", "shuffleAnswers": false, "minAnswers": "4", "variableReplacements": [], "answers": ["{all[select[0]][1]}", "{all[select[1]][1]}", "{all[select[2]][1]}", "{all[select[3]][1]}"], "choices": ["{all[select[0]][0]}", "{all[select[1]][0]}", "{all[select[2]][0]}", "{all[select[3]][0]}"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "maxMarks": 0, "scripts": {}, "marks": 0, "warningType": "warn", "steps": [{"prompt": "<p>Consider the proposition:</p>\n<p><strong>There is someone whose name is not known to the rest of the group.</strong></p>\n<p>Using the predicate $P(m,n)$ we can express this as:</p>\n<p>$\\exists m \\forall n (\\neg P(n,m))$.</p>\n<p>If we negate the proposition then we obtain on using the rules of negating such propositions:</p>\n<p>$\\forall m \\exists n (P(n,m))$.</p>\n<p>This corresponds to the English sentence (not asked for in this question):</p>\n<p><strong>Everyone's name is known by somebody else.</strong></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "showCorrectAnswer": true, "type": "m_n_x", "shuffleChoices": true, "minMarks": 0, "layout": {"expression": "", "type": "all"}}, {"maxAnswers": "4", "variableReplacements": [], "layout": {"expression": "", "type": "all"}, "matrix": "marks", "shuffleAnswers": false, "minAnswers": "4", "shuffleChoices": true, "answers": ["{all[select1[0]][1]}", "{all[select1[1]][1]}", "{all[select1[2]][1]}", "{all[select1[3]][1]}"], "choices": ["{all[select1[0]][0]}", "{all[select1[1]][0]}", "{all[select1[2]][0]}", "{all[select1[3]][0]}"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "maxMarks": 0, "scripts": {}, "marks": 0, "warningType": "warn", "showCorrectAnswer": true, "type": "m_n_x", "minMarks": 0}, {"maxAnswers": "4", "variableReplacements": [], "layout": {"expression": "", "type": "all"}, "matrix": "marks", "shuffleAnswers": false, "minAnswers": "4", "shuffleChoices": true, "answers": ["{all[select2[0]][1]}", "{all[select2[1]][1]}", "{all[select2[2]][1]}", "{all[select2[3]][1]}"], "choices": ["{all[select2[0]][0]}", "{all[select2[1]][0]}", "{all[select2[2]][0]}", "{all[select2[3]][0]}"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "maxMarks": 0, "scripts": {}, "marks": 0, "warningType": "warn", "showCorrectAnswer": true, "type": "m_n_x", "minMarks": 0}], "statement": "<p>In a seminar group we let $P(m,n)$ for group members $m$ and $n$ to be the predicate&nbsp;<strong><em>m knows the name of n&nbsp;</em></strong>.</p>\n<p>Negate each of the following English sentence and choose the corresponding expression for the negated proposition involving quantifiers.</p>\n<p>Note that you will lose one mark for every incorrect choice. However, the minimum mark is $0$.</p>", "variable_groups": [{"variables": ["select"], "name": "Part 0"}, {"variables": ["select1"], "name": "Part 1"}, {"variables": ["select2"], "name": "Part 2"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"all": {"definition": "[['There is someone whose name is not known to the rest of the group.',\n    '$\\\\forall m \\\\exists n (P(n,m))$',\n   'Everybody\\'s name is known by at least one other person.',\n   '$\\\\exists m \\\\forall n (\\\\neg P(n,m))$'],\n  ['Every group member doesn\\'t know the name of at least one other.',\n    '$\\\\exists m \\\\forall n ( P(m,n))$',\n   'Somebody knows the name of everybody.',\n  ' $\\\\forall m \\\\exists n (\\\\neg P(m,n))$'],\n  ['Nobody knows the name of anybody else.',\n    '$\\\\exists m \\\\exists n (P(m,n))$',\n   'At least one person knows the name of another.',\n   '$\\\\forall m \\\\forall n (\\\\neg P(m,n))$'],\n  ['There is a pair of group members who do not know each other\\'s name.',\n    '$\\\\forall m \\\\forall n (P(m,n) \\\\lor P(n,m))$',\n    'Given any pair of members, then at least one of them knows the name of the other.',\n    '$\\\\exists m \\\\exists n (\\\\neg P(m,n) \\\\land \\\\neg P(n,m)$'],\n  ['There is someone who knows everyone\\'s name.',\n    '$\\\\forall m \\\\exists n (\\\\neg P(m,n))$',\n    'Everybody doesn\\'t know the name of at least one other.',\n    '$\\\\exists m \\\\forall n (P(m,n))$'],\n  ['There is at least one person who knows the name of somebody else.',\n    '$\\\\forall m \\\\forall n ( \\\\neg P(n,m))$',\n    'Nobody knows the name of anyone else.',\n    '$\\\\exists m \\\\exists n (P(n,m))$'],\n  ['There is someone who doesn\\'t know the name of at least one other group member.',\n    '$\\\\forall m \\\\forall n (P(n,m))$',\n    'Everybody knows everybody else\\'s names.',\n    '$\\\\exists m \\\\exists n (\\\\neg P(n,m))$'],\n  ['Someone\\'s name is known to everyone else.',\n    '$\\\\forall m \\\\exists n ( \\\\neg P(n,m))$',\n    'Everybody\\'s name is not known by at least one other person.',\n    '$\\\\exists m \\\\forall n ( P(n,m))$'],\n  ['There is at least one person who does not know the name of anybody else.',\n    '$\\\\forall m \\\\exists n ( P(m,n))$',\n    'Everybody knows the name of at least one other.',\n    '$\\\\exists m \\\\forall n (\\\\neg P(m,n))$'],\n  ['Everybody knows at least one other person\\'s name.',\n    '$\\\\exists m \\\\forall n (\\\\neg P(m,n))$',\n    'There is somebody who does not know anyone else\\'s name.',\n    '$\\\\forall m \\\\exists n (P(m,n))$'],\n  ['Any member of the group has at least one person who doesn\\'t know their name.',\n    '$\\\\exists n \\\\forall m (P(m,n))$',\n    'There is someone whose name is known by everyone.',\n    '$\\\\forall n \\\\exists m (\\\\neg P(m,n))$'],\n  ['There are at least two people who know each other\\'s name.',\n    '$\\\\forall n \\\\forall m (\\\\neg P(n,m) \\\\lor \\\\neg P(m,n))$',\n    'For any two people in the group at least one doesn\\'t know the name of the other.',\n    '$\\\\exists n \\\\exists m (P(n,m) \\\\land P(m,n))$']\n  \n  ]", "templateType": "anything", "group": "Ungrouped variables", "name": "all", "description": ""}, "select": {"definition": "shuffle(list(0..length(all)-1))[0..4]", "templateType": "anything", "group": "Part 0", "name": "select", "description": ""}, "select2": {"definition": "list(set(0..length(all)-1)-(set(select) or set(select1)))", "templateType": "anything", "group": "Part 2", "name": "select2", "description": ""}, "select1": {"definition": "list(set(0..length(all)-1)-set(select))[0..4]", "templateType": "anything", "group": "Part 1", "name": "select1", "description": ""}, "marks": {"definition": "list(2*id(4)-matrix(repeat(repeat(1,4),4)))", "templateType": "anything", "group": "Ungrouped variables", "name": "marks", "description": ""}}, "metadata": {"notes": "", "description": "<p>English sentences which are propositions are given&nbsp;and the appropriate logical expression chosen for the negation of the sentence.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}