// Numbas version: exam_results_page_options {"navigation": {"reverse": true, "preventleave": true, "onleave": {"message": "", "action": "none"}, "showresultspage": "oncompletion", "showfrontpage": true, "browse": true, "allowregen": true}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Questions on using quantifiers.

"}, "showQuestionGroupNames": false, "name": "Luis's copy of Quantifiers", "feedback": {"intro": "", "showactualmark": true, "advicethreshold": 0, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "feedbackmessages": []}, "percentPass": 0, "timing": {"allowPause": true, "timeout": {"message": "", "action": "none"}, "timedwarning": {"message": "", "action": "none"}}, "showstudentname": true, "question_groups": [{"pickingStrategy": "all-shuffled", "name": "Group", "pickQuestions": 1, "questions": [{"name": "Quantifiers 2-", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "tags": ["logic", "quantifiers", "statements"], "metadata": {"description": "

English sentences which are propositions are given and for each the appropriate proposition involving quantifiers is to be chosen.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

In a seminar group, for group members $m$ and $n$, we let $P(m,n)$ to be the predicate m knows the name of n .

For each English sentence choose the corresponding proposition involving quantifiers.

Note that you will lose one mark for every incorrect choice. However, the minimum mark is $0$.

", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"select": {"name": "select", "group": "Part 0", "definition": "shuffle(list(0..length(all)-1))[0..4]", "description": "", "templateType": "anything", "can_override": false}, "select2": {"name": "select2", "group": "Part 2", "definition": "list(set(0..length(all)-1)-(set(select) or set(select1)))", "description": "", "templateType": "anything", "can_override": false}, "select1": {"name": "select1", "group": "Part 1", "definition": "list(set(0..length(all)-1)-set(select))[0..4]", "description": "", "templateType": "anything", "can_override": false}, "marks_matrix": {"name": "marks_matrix", "group": "Ungrouped variables", "definition": "matrix(list(2*id(4)-matrix(repeat(repeat(1,4),4))))", "description": "", "templateType": "anything", "can_override": false}, "all": {"name": "all", "group": "Ungrouped variables", "definition": "[['There is someone not known to the rest.',\n '$\\\\exists m \\\\forall n (\\\\neg P(n,m))$'],\n ['Every group member doesn\\'t know the name of at least one other.',\n '$\\\\forall m \\\\exists n (\\\\neg P(m,n))$'],\n ['Nobody knows the name of anybody else.',\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 '$\\\\exists m \\\\exists n (\\\\neg P(m,n) \\\\land \\\\neg P(n,m)$'],\n ['There is someone who knows everyone\\'s name.',\n '$\\\\exists m \\\\forall n (P(m,n))$'],\n ['There is at least one person who knows the name of somebody 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 '$\\\\exists m \\\\exists n (\\\\neg P(n,m))$'],\n ['Someone\\'s name is known to everyone else.',\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 '$\\\\exists m \\\\forall n (\\\\neg P(m,n))$'],\n ['Everybody knows at least one other person\\'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 '$\\\\forall n \\\\exists m (\\\\neg P(m,n))$'],\n ['There are at least two people who know each other\\'s name.',\n '$\\\\exists n \\\\exists m (P(n,m) \\\\land P(m,n))$']\n \n ]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["all", "marks_matrix"], "variable_groups": [{"name": "Part 0", "variables": ["select"]}, {"name": "Part 1", "variables": ["select1"]}, {"name": "Part 2", "variables": ["select2"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_x", "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": "

The numbers heading the columns refer to the following:

1. {all[select[0]][1]}

2. {all[select[1]][1]}

3. {all[select[2]][1]}

4. {all[select[3]][1]}

", "minMarks": 0, "maxMarks": 0, "minAnswers": "4", "maxAnswers": "4", "shuffleChoices": true, "shuffleAnswers": false, "displayType": "radiogroup", "warningType": "warn", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["{all[select[0]][0]}", "{all[select[1]][0]}", "{all[select[2]][0]}", "{all[select[3]][0]}"], "matrix": "marks_matrix", "layout": {"type": "all", "expression": ""}, "answers": ["

", "

"]}, {"type": "m_n_x", "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": "

The numbers heading the columns refer to the following:

1. {all[select1[0]][1]}

2. {all[select1[1]][1]}

3. {all[select1[2]][1]}

4. {all[select1[3]][1]}

", "minMarks": 0, "maxMarks": 0, "minAnswers": "4", "maxAnswers": "4", "shuffleChoices": true, "shuffleAnswers": false, "displayType": "radiogroup", "warningType": "warn", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["{all[select1[0]][0]}", "{all[select1[1]][0]}", "{all[select1[2]][0]}", "{all[select1[3]][0]}"], "matrix": "marks_matrix", "layout": {"type": "all", "expression": ""}, "answers": ["

", "

The numbers heading the columns refer to the following:

1. {all[select2[0]][1]}

2. {all[select2[1]][1]}

3. {all[select2[2]][1]}

4. {all[select2[3]][1]}

", "minMarks": 0, "maxMarks": 0, "minAnswers": "4", "maxAnswers": "4", "shuffleChoices": true, "shuffleAnswers": false, "displayType": "radiogroup", "warningType": "warn", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["{all[select2[0]][0]}", "{all[select2[1]][0]}", "{all[select2[2]][0]}", "{all[select2[3]][0]}"], "matrix": "marks_matrix", "layout": {"type": "all", "expression": ""}, "answers": ["

", "

"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Quantifiers 3-", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "tags": ["logic", "quantifiers", "statements"], "metadata": {"description": "

English sentences which are propositions are given and the appropriate logical expression chosen for the negation of the sentence.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

In a seminar group, for group members $m$ and $n$, we let $P(m,n)$ be the predicate m knows the name of n .

Negate each of the following English sentences and choose the corresponding expression for the negated proposition involving quantifiers.

Note that you will lose one mark for every incorrect choice. However, the minimum mark is $0$.

", "advice": "

In the following we use the rules for negating a proposition involving predicates and quantifiers as given in the lectures.

1. The sentence:

{all[select[0]][0]}

can be written in predicate form with quantifiers as:

{all[select[0]][3]}

The negation of the sentence can be written as:

{all[select[0]][2]}

The predicate form with quantifiers for this is:

{all[select[0]][1]}

2. The sentence:

{all[select[1]][0]}

can be written in predicate form with quantifiers as:

{all[select[1]][3]}

The negation of the sentence can be written as:

{all[select[1]][2]}

The predicate form with quantifiers for this is:

{all[select[1]][1]}

3. The sentence:

{all[select[2]][0]}

can be written in predicate form with quantifiers as:

{all[select[2]][3]}

The negation of the sentence can be written as:

{all[select[2]][2]}

The predicate form with quantifiers for this is:

{all[select[2]][1]}

4. The sentence:

{all[select[3]][0]}

can be written in predicate form with quantifiers as:

{all[select[3]][3]}

The negation of the sentence can be written as:

{all[select[3]][2]}

The predicate form with quantifiers for this is:

{all[select[3]][1]}

Similarly for Parts b) and c).

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"all": {"name": "all", "group": "Ungrouped variables", "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 ]", "description": "", "templateType": "anything", "can_override": false}, "select2": {"name": "select2", "group": "Part 2", "definition": "list(set(0..length(all)-1)-(set(select) or set(select1)))", "description": "", "templateType": "anything", "can_override": false}, "select1": {"name": "select1", "group": "Part 1", "definition": "list(set(0..length(all)-1)-set(select))[0..4]", "description": "", "templateType": "anything", "can_override": false}, "marks_matrix": {"name": "marks_matrix", "group": "Ungrouped variables", "definition": "list(2*id(4)-matrix(repeat(repeat(1,4),4)))", "description": "", "templateType": "anything", "can_override": false}, "select": {"name": "select", "group": "Part 0", "definition": "shuffle(list(0..length(all)-1))[0..4]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["all", "marks_matrix"], "variable_groups": [{"name": "Part 0", "variables": ["select"]}, {"name": "Part 1", "variables": ["select1"]}, {"name": "Part 2", "variables": ["select2"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_x", "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": "

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.

", "stepsPenalty": "1", "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": "

Consider the proposition:

There is someone whose name is not known to the rest of the group.

Using the predicate $P(m,n)$ we can express this as:

$\\exists m \\forall n (\\neg P(n,m))$.

If we negate the proposition then we obtain on using the rules of negating such propositions:

$\\forall m \\exists n (P(n,m))$.

This corresponds to the English sentence (not asked for in this question):

Everyone's name is known by somebody else.

"}], "minMarks": 0, "maxMarks": 0, "minAnswers": "4", "maxAnswers": "4", "shuffleChoices": true, "shuffleAnswers": false, "displayType": "radiogroup", "warningType": "warn", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["{all[select[0]][0]}", "{all[select[1]][0]}", "{all[select[2]][0]}", "{all[select[3]][0]}"], "matrix": "marks_matrix", "layout": {"type": "all", "expression": ""}, "answers": ["{all[select[0]][1]}", "{all[select[1]][1]}", "{all[select[2]][1]}", "{all[select[3]][1]}"]}, {"type": "m_n_x", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "minAnswers": "4", "maxAnswers": "4", "shuffleChoices": true, "shuffleAnswers": false, "displayType": "radiogroup", "warningType": "warn", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["{all[select1[0]][0]}", "{all[select1[1]][0]}", "{all[select1[2]][0]}", "{all[select1[3]][0]}"], "matrix": "marks_matrix", "layout": {"type": "all", "expression": ""}, "answers": ["{all[select1[0]][1]}", "{all[select1[1]][1]}", "{all[select1[2]][1]}", "{all[select1[3]][1]}"]}, {"type": "m_n_x", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "minAnswers": "4", "maxAnswers": "4", "shuffleChoices": true, "shuffleAnswers": false, "displayType": "radiogroup", "warningType": "warn", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["{all[select2[0]][0]}", "{all[select2[1]][0]}", "{all[select2[2]][0]}", "{all[select2[3]][0]}"], "matrix": "marks_matrix", "layout": {"type": "all", "expression": ""}, "answers": ["{all[select2[0]][1]}", "{all[select2[1]][1]}", "{all[select2[2]][1]}", "{all[select2[3]][1]}"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Quantifiers 4--AJD", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"Prop_const": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[random(Qchoice),random(Qchoice),random(Qchoice),random(Achoice),random(Bchoice),random(Cchoice)]", "description": "", "name": "Prop_const"}, "wrong_flags": {"templateType": "anything", "group": "Ungrouped variables", "definition": "shuffle([shuffle([0,1,1,1,1]),shuffle([0,0,1,1,1]),shuffle([0,0,0,1,1]),shuffle([0,0,0,0,1])])[0..4]", "description": "", "name": "wrong_flags"}, "prop2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[random(Achoice),random(Bchoice)]", "description": "", "name": "prop2"}, "Cchoice": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[[\"\\\\rightarrow\",\"\\\\land\"],\n [\"\\\\land\",\"\\\\rightarrow\"]\n ]", "description": "", "name": "Cchoice"}, "Achoice": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[\n[\"c \\\\ge b\"],\n[\"c < b\"],\n[\"c = b\"],\n[\"c \\\\neq b\"],\n[\"2c\\\\le b\"],\n[\"c\\\\ge b+1\"],\n[\"c < b+1\"],\n[\"2c > b\"]\n]", "description": "", "name": "Achoice"}, "Qchoice": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[\n [\"\\\\forall\",\"\\\\exists\"],\n [\"\\\\exists\",\"\\\\forall\"]\n]", "description": "", "name": "Qchoice"}, "Bchoice": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[\n[\"|(-1)\\^c| < a\",\"|(-1)\\^c| \\\\ge a\"],\n[\"\\\\left|\\\\sqrt\\{\\\\left(1-\\\\frac\\{1\\}\\{c\\}\\\\right)\\}-1\\\\right| < a\",\"a \\\\leq \\\\left| \\\\sqrt\\{\\\\left(1-\\\\frac\\{1\\}\\{c\\}\\\\right)\\}-1 \\\\right|\"],\n[\"\\\\frac\\{1\\}\\{2\\^c\\} < 3a\",\"\\\\frac\\{1\\}\\{2\\^c\\} \\\\ge 3a\"],\n[\"\\\\frac\\{c\\^2-2\\}\\{2c+3\\} > 3a+6\",\"\\\\frac\\{c\\^2-2\\}\\{2c+3\\} \\\\le 3a+6\"],\n[\"\\\\left|\\\\frac\\{1\\}\\{c\\^3\\}\\\\right| < a\",\"\\\\left|\\\\frac\\{1\\}\\{c\\^3\\}\\\\right| \\\\ge a\"],\n[\"\\\\frac\\{c\\^2+6\\}\\{2c\\^2+1\\} < a\",\"\\\\frac\\{c\\^2+6\\}\\{2c\\^2+1\\} \\\\ge a\"]\n]\n", "description": "", "name": "Bchoice"}}, "ungrouped_variables": ["Prop_const", "Cchoice", "Achoice", "Qchoice", "Bchoice", "wrong_flags", "prop2"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"displayType": "radiogroup", "choices": ["

$\\displaystyle \\var{latex(Prop_const[0][1])} a \\in \\mathbb{R}^+, \\var{latex(Prop_const[1][1])} b \\in \\mathbb{N}, \\var{latex(Prop_const[2][1])} c \\in \\mathbb{N}\\,\\left[(\\var{latex(Prop_const[3][0])})\\, \\var{latex(Prop_const[5][1])}\\, \\left(\\var{latex(Prop_const[4][1])}\\right)\\right]$

", "

$\\displaystyle \\var{latex(Prop_const[0][wrong_flags[0][0]])} a \\in \\mathbb{R}^+, \\var{latex(Prop_const[1][wrong_flags[0][1]])} b \\in \\mathbb{N}, \\var{latex(Prop_const[2][wrong_flags[0][2]])} c \\in \\mathbb{N}\\,\\left[(\\var{latex(Prop_const[3][0])})\\, \\var{latex(Prop_const[5][wrong_flags[0][3]])}\\, \\left(\\var{latex(Prop_const[4][wrong_flags[0][4]])}\\right)\\right]$

", "

$\\displaystyle \\var{latex(Prop_const[0][wrong_flags[1][0]])} a \\in \\mathbb{R}^+, \\var{latex(Prop_const[1][wrong_flags[1][1]])} b \\in \\mathbb{N}, \\var{latex(Prop_const[2][wrong_flags[1][2]])} c \\in \\mathbb{N}\\,\\left[(\\var{latex(Prop_const[3][0])})\\, \\var{latex(Prop_const[5][wrong_flags[1][3]])}\\, \\left(\\var{latex(Prop_const[4][wrong_flags[1][4]])}\\right)\\right]$

", "

$\\displaystyle \\var{latex(Prop_const[0][wrong_flags[2][0]])} a \\in \\mathbb{R}^+, \\var{latex(Prop_const[1][wrong_flags[2][1]])} b \\in \\mathbb{N}, \\var{latex(Prop_const[2][wrong_flags[2][2]])} c \\in \\mathbb{N}\\,\\left[(\\var{latex(Prop_const[3][0])})\\, \\var{latex(Prop_const[5][wrong_flags[2][3]])}\\, \\left(\\var{latex(Prop_const[4][wrong_flags[2][4]])}\\right)\\right]$

"], "showCorrectAnswer": true, "displayColumns": "1", "prompt": "

Choose the negation of \\[\\var{latex(Prop_const[0][0])} a \\in \\mathbb{R}^+, \\var{latex(Prop_const[1][0])} b \\in \\mathbb{N}, \\var{latex(Prop_const[2][0])} c \\in \\mathbb{N}\\,\\left[(\\var{latex(Prop_const[3][0])})\\, \\var{latex(Prop_const[5][0])}\\, \\left(\\var{latex(Prop_const[4][0])}\\right)\\right]\\]

from the list below

", "distractors": ["", "", "", ""], "variableReplacements": [], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": "4", "variableReplacementStrategy": "originalfirst", "matrix": ["4", 0, 0, 0], "marks": 0}, {"displayType": "radiogroup", "choices": ["

$\\displaystyle \\var{latex(Prop_const[0][0])} a \\in \\mathbb{R}^+, \\var{latex(Prop_const[1][1])} b \\in \\mathbb{N}, \\var{latex(Prop_const[2][0])} c \\in \\mathbb{N}\\,\\left[(\\var{latex(prop2[0][0])})\\, \\var{latex(Prop_const[5][0])}\\, \\left(\\var{latex(prop2[1][0])}\\right)\\right]$

", "

$\\displaystyle \\var{latex(Prop_const[0][1-wrong_flags[1][0]])} a \\in \\mathbb{R}^+, \\var{latex(Prop_const[1][wrong_flags[1][1]])} b \\in \\mathbb{N}, \\var{latex(Prop_const[2][1-wrong_flags[1][2]])} c \\in \\mathbb{N}\\,\\left[(\\var{latex(prop2[0][0])})\\, \\var{latex(Prop_const[5][1-wrong_flags[1][3]])}\\, \\left(\\var{latex(prop2[1][1-wrong_flags[1][4]])}\\right)\\right]$

", "

$\\displaystyle \\var{latex(Prop_const[0][1-wrong_flags[2][0]])} a \\in \\mathbb{R}^+, \\var{latex(Prop_const[1][wrong_flags[2][1]])} b \\in \\mathbb{N}, \\var{latex(Prop_const[2][1-wrong_flags[2][2]])} c \\in \\mathbb{N}\\,\\left[(\\var{latex(prop2[0][0])})\\, \\var{latex(Prop_const[5][1-wrong_flags[2][3]])}\\, \\left(\\var{latex(prop2[1][1-wrong_flags[2][4]])}\\right)\\right]$

", "

$\\displaystyle \\var{latex(Prop_const[0][1-wrong_flags[3][0]])} a \\in \\mathbb{R}^+, \\var{latex(Prop_const[1][wrong_flags[3][1]])} b \\in \\mathbb{N}, \\var{latex(Prop_const[2][1-wrong_flags[3][2]])} c \\in \\mathbb{N}\\,\\left[(\\var{latex(prop2[0][0])})\\, \\var{latex(Prop_const[5][1-wrong_flags[3][3]])}\\, \\left(\\var{latex(prop2[1][1-wrong_flags[3][4]])}\\right)\\right]$

"], "showCorrectAnswer": true, "displayColumns": "1", "prompt": "

Choose the negation of

\\[\\var{latex(Prop_const[0][1])} a \\in \\mathbb{R}^+, \\var{latex(Prop_const[1][0])} b \\in \\mathbb{N}, \\var{latex(Prop_const[2][1])} c \\in \\mathbb{N}\\,\\left[(\\var{latex(prop2[0][0])})\\, \\var{latex(Prop_const[5][1])}\\, \\left(\\var{latex(prop2[1][1])}\\right)\\right]\\]

from the list below

", "tags": ["exists", "for all", "logic", "logical expressions", "negation of logical expressions", "negation of quantifiers", "predicates", "quantifiers"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "

Given two propositions in mathematics using quantifiers, choose the corresponding negation of the proposition. For example, the negation of: $\\displaystyle \\exists a \\in \\mathbb{R^+},\\;\\exists b \\in \\mathbb{N},\\;\\exists c \\in \\mathbb{N}\\;\\left[(c \\lt b+1) \\land \\left(\\frac{1}{2^n} \\geq 3a\\right)\\right]$

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": ""}, {"name": "Quantifiers1-", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [{"variables": ["marking_matrix", "select"], "name": "Part 0"}, {"variables": ["select1", "marking_matrix1"], "name": "Part 1"}, {"variables": ["select2", "marking_matrix2"], "name": "Part 2"}], "variables": {"neg_marks": {"group": "Ungrouped variables", "templateType": "anything", "definition": "2*id(3)+matrix(map(map(-1,x,0..2),y,0..2))", "name": "neg_marks", "description": ""}, "marking_matrix1": {"group": "Part 1", "templateType": "anything", "definition": "map(list(neg_marks[x])+[all[select1[x]][2]]+[-1*all[select1[x]][2]],x,0..2)", "name": "marking_matrix1", "description": ""}, "marking_matrix": {"group": "Part 0", "templateType": "anything", "definition": "map(list(neg_marks[x])+[all[select[x]][2]]+[-1*all[select[x]][2]],x,0..2)", "name": "marking_matrix", "description": ""}, "select1": {"group": "Part 1", "templateType": "anything", "definition": "list(set(0..length(all)-1)-set(select))[0..3]", "name": "select1", "description": ""}, "select": {"group": "Part 0", "templateType": "anything", "definition": "shuffle(list(0..length(all)-1))[0..3]", "name": "select", "description": ""}, "all": {"group": "Ungrouped variables", "templateType": "anything", "definition": "[['The square of any real number is greater than $0$.',\n '$\\\\forall x \\\\in \\\\mathbb{R}\\\\;(x^2\\\\gt 0).$',-1],\n ['Given a real number then some integral power is not negative.',\n '$\\\\forall x \\\\in \\\\mathbb{R} \\\\;\\\\exists n \\\\in \\\\mathbb{N}\\\\;(x^n\\\\geq 0).$',1],\n ['A subset of the natural numbers is a subset of the reals.',\n '$\\\\forall X \\\\subseteq \\\\mathbb{N}\\\\;(X\\\\subseteq \\\\mathbb{R}).$',1],\n ['For every natural number $n$ there is a subset of $\\\\mathbb{N}$ with less than $n$ members.',\n '$\\\\forall n \\\\in \\\\mathbb{N}\\\\;\\\\exists X \\\\subseteq \\\\mathbb{N}\\\\;(|X|\\\\lt n).$',1],\n ['All subsets of the natural numbers have less than a fixed number of elements.',\n '$\\\\exists n \\\\in \\\\mathbb{N}\\\\; \\\\forall X \\\\subseteq \\\\mathbb{N}\\\\;(|X|\\\\lt n).$',-1],\n ['All subsets of the natural numbers are finite.',\n '$\\\\forall X \\\\subseteq \\\\mathbb{N}\\\\;\\\\exists n \\\\in \\\\mathbb{Z}\\\\;(|X|=n).$',-1],\n ['Given an integer $n$, there is a subset of the natural numbers with $n$ elements.',\n '$\\\\forall n \\\\in \\\\mathbb{Z}\\\\;\\\\exists X \\\\subseteq \\\\mathbb{N}\\\\;(|X|=n).$',-1],\n ['Given an integer, then adding $5$ to it gives another integer.',\n '$\\\\forall n \\\\in \\\\mathbb{Z}\\\\; \\\\exists m \\\\in \\\\mathbb{Z}\\\\; (m=n+5).$',1],\n ['There is an integer $t$ such that adding $5$ to any integer gives $t$.',\n '$\\\\exists m \\\\in \\\\mathbb{Z}\\\\; \\\\forall n \\\\in \\\\mathbb{Z}\\\\; (m=n+5).$',-1]\n ]", "name": "all", "description": ""}, "select2": {"group": "Part 2", "templateType": "anything", "definition": "list(set(0..length(all)-1)-(set(select) or set(select1)))", "name": "select2", "description": ""}, "marking_matrix2": {"group": "Part 2", "templateType": "anything", "definition": "map(list(neg_marks[x])+[all[select2[x]][2]]+[-1*all[select2[x]][2]],x,0..2)", "name": "marking_matrix2", "description": ""}}, "ungrouped_variables": ["all", "neg_marks"], "rulesets": {}, "showQuestionGroupNames": false, "functions": {}, "parts": [{"displayType": "checkbox", "layout": {"type": "all", "expression": ""}, "choices": ["{all[select[0]][0]}", "{all[select[1]][0]}", "{all[select[2]][0]}"], "showCorrectAnswer": true, "matrix": "marking_matrix", "minAnswers": "6", "maxAnswers": "6", "shuffleChoices": true, "warningType": "warn", "scripts": {}, "minMarks": 0, "type": "m_n_x", "maxMarks": 0, "shuffleAnswers": false, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "answers": ["{all[select[0]][1]}", "{all[select[1]][1]}", "{all[select[2]][1]}", "True", "False"]}, {"displayType": "checkbox", "layout": {"type": "all", "expression": ""}, "choices": ["{all[select1[0]][0]}", "{all[select1[1]][0]}", "{all[select1[2]][0]}"], "showCorrectAnswer": true, "matrix": "marking_matrix1", "minAnswers": "6", "maxAnswers": "6", "shuffleChoices": true, "warningType": "warn", "scripts": {}, "minMarks": 0, "type": "m_n_x", "maxMarks": 0, "shuffleAnswers": false, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "answers": ["{all[select1[0]][1]}", "{all[select1[1]][1]}", "{all[select1[2]][1]}", "True", "False"]}, {"displayType": "checkbox", "layout": {"type": "all", "expression": ""}, "choices": ["{all[select2[0]][0]}", "{all[select2[1]][0]}", "{all[select2[2]][0]}"], "showCorrectAnswer": true, "matrix": "marking_matrix2", "minAnswers": "6", "maxAnswers": "6", "shuffleChoices": true, "warningType": "warn", "scripts": {}, "minMarks": 0, "type": "m_n_x", "maxMarks": 0, "shuffleAnswers": false, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "answers": ["{all[select2[0]][1]}", "{all[select2[1]][1]}", "{all[select2[2]][1]}", "True", "False"]}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

Choose the appropriate proposition for the following English sentences. Also choose whether they are true or false.

You must make $2$ choices in each row, one of which is to determine whether the proposition is true or false.

Note also that every wrong answer takes away one from your score. However, your minimum score is $0$.

", "tags": ["logic", "quantifiers", "statements"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "

English sentences are given and for each the appropriate proposition involving quantifiers is to be chosen. Also choose whether the propositions are true or false.

"}, "advice": ""}]}], "duration": 0, "type": "exam", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}], "extensions": [], "custom_part_types": [], "resources": []}