// Numbas version: finer_feedback_settings
{"name": "Daniel's copy of Quantifiers1-", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"js": "", "css": ""}, "variable_groups": [{"name": "Part 0", "variables": ["marking_matrix", "select"]}, {"name": "Part 1", "variables": ["select1", "marking_matrix1"]}, {"name": "Part 2", "variables": ["select2", "marking_matrix2"]}], "ungrouped_variables": ["all", "neg_marks"], "metadata": {"description": "<p>English sentences are given and for each the appropriate proposition&nbsp;involving quantifiers is to be chosen. Also choose whether the propositions are true or false.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "extensions": [], "variables": {"select": {"templateType": "anything", "name": "select", "definition": "shuffle(list(0..length(all)-1))[0..3]", "group": "Part 0", "description": ""}, "marking_matrix": {"templateType": "anything", "name": "marking_matrix", "definition": "map(list(neg_marks[x])+[all[select[x]][2]]+[-1*all[select[x]][2]],x,0..2)", "group": "Part 0", "description": ""}, "select1": {"templateType": "anything", "name": "select1", "definition": "list(set(0..length(all)-1)-set(select))[0..3]", "group": "Part 1", "description": ""}, "all": {"templateType": "anything", "name": "all", "definition": "[['The square of any real number is greater than $0$.',\n    '$\\\\forall x \\\\in \\\\mathbb{R}\\\\\\\\ \\\\;(x^2\\\\gt 0).$',-0.5],\n  ['Any real number can become non-negative when raised to some integer power.',\n    '$\\\\forall x \\\\in \\\\mathbb{R} \\\\;\\\\exists n \\\\in \\\\mathbb{N}\\\\\\\\ \\\\;(x^n\\\\geq 0).$',0.5],\n  ['A subset of the natural numbers is a subset of the reals.',\n    '$\\\\forall X \\\\subseteq \\\\mathbb{N}\\\\\\\\ \\\\;(X\\\\subseteq \\\\mathbb{R}).$',0.5],\n  ['For every natural number $n$ there is a subset of $\\\\mathbb{N}$ with less than $n$ members.',\n    '$\\\\forall n \\\\in \\\\mathbb{N}\\\\;\\\\exists X \\\\subseteq \\\\mathbb{N}\\\\\\\\ \\\\;(|X|\\\\lt n).$',-0.5],\n  ['All subsets of the natural numbers have less than a fixed number of elements.',\n    '$\\\\exists n \\\\in \\\\mathbb{N}\\\\; \\\\forall X \\\\subseteq \\\\mathbb{N}\\\\\\\\ \\\\;(|X|\\\\lt n).$',-0.5],\n  ['All subsets of the natural numbers are finite.',\n    '$\\\\forall X \\\\subseteq \\\\mathbb{N}\\\\;\\\\exists n \\\\in \\\\mathbb{Z}\\\\\\\\ \\\\;(|X|=n).$',-0.5],\n  ['Given an integer $n$, there is a subset of the natural numbers with $n$ elements.',\n    '$\\\\forall n \\\\in \\\\mathbb{Z}\\\\;\\\\exists X \\\\subseteq \\\\mathbb{N}\\\\\\\\ \\\\;(|X|=n).$',-0.5],\n  ['Given an integer, then adding $5$ to it gives another integer.',\n    '$\\\\forall n \\\\in \\\\mathbb{Z}\\\\; \\\\exists m \\\\in \\\\mathbb{Z}\\\\\\\\ \\\\; (m=n+5).$',0.5],\n  ['There is an integer $m$ such that adding $5$ to any integer gives $m$.',\n    '$\\\\exists m \\\\in \\\\mathbb{Z}\\\\; \\\\forall n \\\\in \\\\mathbb{Z}\\\\\\\\ \\\\; (m=n+5).$',-0.5]\n  ]", "group": "Ungrouped variables", "description": ""}, "marking_matrix1": {"templateType": "anything", "name": "marking_matrix1", "definition": "map(list(neg_marks[x])+[all[select1[x]][2]]+[-1*all[select1[x]][2]],x,0..2)", "group": "Part 1", "description": ""}, "select2": {"templateType": "anything", "name": "select2", "definition": "list(set(0..length(all)-1)-(set(select) or set(select1)))", "group": "Part 2", "description": ""}, "neg_marks": {"templateType": "anything", "name": "neg_marks", "definition": "id(3)+matrix(map(map(-0.5,x,0..2),y,0..2))", "group": "Ungrouped variables", "description": ""}, "marking_matrix2": {"templateType": "anything", "name": "marking_matrix2", "definition": "map(list(neg_marks[x])+[all[select2[x]][2]]+[-1*all[select2[x]][2]],x,0..2)", "group": "Part 2", "description": ""}}, "advice": "<p>$\\forall x \\in \\mathbb{R}\\;(x^2\\gt 0)$ is false, because there is a single counterexample when $x=0$ (in which case $x^2=0$, which is not allowed in the strict inequality).</p>\n<p>$\\forall x \\in \\mathbb{R} \\;\\exists n \\in \\mathbb{N}\\;(x^n\\geq 0)$ is true, because it is always true for any even value of $n$. As seen in the previous example, $n=2$ will work here because the inequality is no longer strict. You might also consider the simpler case $n=0$, though $x^0$ is not well-defined when $x=0$.</p>\n<p>$\\forall X \\subseteq \\mathbb{N}\\;(X\\subseteq \\mathbb{R})$ is true, because $\\mathbb{N} \\subset \\mathbb{R}$ itself - that is, all natural numbers are also real numbers.</p>\n<p>$\\forall n \\in \\mathbb{N}\\;\\exists X \\subseteq \\mathbb{N}\\;(|X|\\lt n)$ is false, because there is a single counterexample when $n=0$ (in which case $|X|$ cannot possibly be negative). Notice $n=1$ is not a counterexample since $|\\emptyset| = 0 &lt; 1$.</p>\n<p>$\\exists n \\in \\mathbb{N}\\; \\forall X \\subseteq \\mathbb{N}\\;(|X|\\lt n)$ is false, because for any given natural number $n$, we could take $X$ to be the set $\\{0,1,2,\\ldots,n\\}$, which has size $n+1$ and so $|X| \\nless n$.</p>\n<p>$\\forall X \\subseteq \\mathbb{N}\\;\\exists n \\in \\mathbb{Z}\\;(|X|=n)$ is false, because there exist subsets of $\\mathbb{N}$ which are infinite in size, and therefore cannot have cardinality equal to a natural number. For example, $\\mathbb{N}$ itself is an infinite subset of $\\mathbb{N}$, as is, for example, the set of all even natural numbers $\\{2k \\; | \\; k \\in \\mathbb{N}\\}$.</p>\n<p>$\\forall n \\in \\mathbb{Z}\\;\\exists X \\subseteq \\mathbb{N}\\;(|X|=n)$ is false, because the statement breaks for any negative value of $n$, which is allowed since $\\mathbb{Z}$ includes the negative integers. No set can have negative cardinality.&nbsp;</p>\n<p>$\\forall n \\in \\mathbb{Z}\\; \\exists m \\in \\mathbb{Z}\\; (m=n+5)$ is true, because $\\mathbb{Z}$ is closed under addition, and $5 \\in \\mathbb{Z}$. That is, the set of integers has the property that adding any two integers will always return an integer.</p>\n<p>$\\exists m \\in \\mathbb{Z}\\; \\forall n \\in \\mathbb{Z}\\; (m=n+5)$ is false, because any two different values of $n$ will yield a different $m$ (e.g. $n=0 \\implies m=5$, but $n=1 \\implies m=6$), so $m=n+5$ for a fixed $m$ cannot hold for even two different values of $n$, let alone all integer values.</p>", "tags": [], "statement": "<p>Choose the appropriate proposition for the following English sentences. Also choose whether they are true or false.</p>\n<p>You must choose two of the five options presented in each row. You may need to scroll over to see all five options.</p>\n<p>Note also that every wrong answer takes away from your score. However, your minimum score is $0$.</p>", "parts": [{"displayType": "checkbox", "layout": {"type": "all", "expression": ""}, "answers": ["{all[select[0]][1]}", "{all[select[1]][1]}", "{all[select[2]][1]}", "True", "False"], "maxMarks": 0, "scripts": {}, "showCorrectAnswer": true, "unitTests": [], "showCellAnswerState": true, "choices": ["{all[select[0]][0]}", "{all[select[1]][0]}", "{all[select[2]][0]}"], "minAnswers": "0", "shuffleAnswers": false, "customMarkingAlgorithm": "", "minMarks": 0, "matrix": "marking_matrix", "variableReplacementStrategy": "originalfirst", "useCustomName": false, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacements": [], "shuffleChoices": true, "warningType": "warn", "marks": 0, "customName": "", "type": "m_n_x", "maxAnswers": "6"}, {"displayType": "checkbox", "layout": {"type": "all", "expression": ""}, "answers": ["{all[select1[0]][1]}", "{all[select1[1]][1]}", "{all[select1[2]][1]}", "True", "False"], "maxMarks": 0, "scripts": {}, "showCorrectAnswer": true, "unitTests": [], "showCellAnswerState": true, "choices": ["{all[select1[0]][0]}", "{all[select1[1]][0]}", "{all[select1[2]][0]}"], "minAnswers": "0", "shuffleAnswers": false, "customMarkingAlgorithm": "", "minMarks": 0, "matrix": "marking_matrix1", "variableReplacementStrategy": "originalfirst", "useCustomName": false, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacements": [], "shuffleChoices": true, "warningType": "warn", "marks": 0, "customName": "", "type": "m_n_x", "maxAnswers": "6"}, {"displayType": "checkbox", "layout": {"type": "all", "expression": ""}, "answers": ["{all[select2[0]][1]}", "{all[select2[1]][1]}", "{all[select2[2]][1]}", "True", "False"], "maxMarks": 0, "scripts": {}, "showCorrectAnswer": true, "unitTests": [], "showCellAnswerState": true, "choices": ["{all[select2[0]][0]}", "{all[select2[1]][0]}", "{all[select2[2]][0]}"], "minAnswers": "0", "shuffleAnswers": false, "customMarkingAlgorithm": "", "minMarks": 0, "matrix": "marking_matrix2", "variableReplacementStrategy": "originalfirst", "useCustomName": false, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacements": [], "shuffleChoices": true, "warningType": "warn", "marks": 0, "customName": "", "type": "m_n_x", "maxAnswers": "6"}], "rulesets": {}, "functions": {}, "name": "Daniel's copy of Quantifiers1-", "type": "question", "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/"}]}