$\\exists z\\in \\mathbb Z~:~\\forall x\\in \\mathbb Z~:~x\\le z\\le 2 x $
", "$\\forall z\\in \\mathbb Z~:~\\exists x\\in \\mathbb Z~:~x\\le z\\le 2 x $
", "$\\forall x\\in \\mathbb Z~:~\\exists z\\in \\mathbb Z~:~ x\\le z\\le 2 x $
", "$\\exists x\\in \\mathbb Z~:~\\exists z\\in \\mathbb Z~:~x\\le z\\le 2 x $
"], "displayColumns": "1", "minMarks": 0, "showFeedbackIcon": true, "variableReplacements": [], "minAnswers": 0, "showCorrectAnswer": true, "marks": 0, "maxMarks": "1", "distractors": ["Incorrect. The order of the quantifiers $\\forall x~\\exists z$ matters.", "Incorrect. The names of the variables in $\\forall x~\\exists z$ matter.", "", "Incorrect: $\\exists x$ is not the same as $\\forall x$."], "maxAnswers": "4", "matrix": ["-1", "-1", "1", "-1"], "variableReplacementStrategy": "originalfirst", "displayType": "checkbox", "shuffleChoices": true, "scripts": {}, "warningType": "none", "type": "m_n_2", "prompt": "Consider the following Statement 1:
\nFor every integer $x$ there is an integer $z$ such that $x\\le z\\le 2 x$.
\n\nLet us first rewrite this statement using mathematical notation for the sets and quantifiers.
\nWhich of the following is a correct formulation of Statement 1? Choose any that apply.
"}, {"choices": ["yes
", "no
"], "displayColumns": 0, "minMarks": 0, "showFeedbackIcon": true, "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "maxMarks": 0, "distractors": ["For $x=0$ and $z=0$ the statement is true.", "For $x=0$ and $z=0$ the statement is true."], "matrix": ["1", 0], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "shuffleChoices": false, "scripts": {}, "type": "1_n_2", "prompt": "Let $x$ be an integer and consider the following Statement 2:
\n$\\exists z\\in \\mathbb Z~:~ x\\le z\\le 2 x $
\nIs Statement 2 correct if $x=0$?
"}, {"showFeedbackIcon": true, "variableReplacements": [], "minValue": "-19999999999999999", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "marks": "1", "mustBeReduced": false, "type": "numberentry", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "scripts": {}, "allowFractions": false, "maxValue": "-1", "prompt": "Is there some integer $x$ so that Statement 2 in b) is false?
\nIf not, enter $0$, otherwise give such an integer $x$ (you will get additional feedback in part d):
"}, {"choices": ["yes
", "no
"], "displayColumns": 0, "minMarks": 0, "showFeedbackIcon": true, "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "maxMarks": 0, "distractors": ["No, statement 1 is incorrect. If $x\\le 2x$, then a suitable $z$ so that $x\\le z\\le 2x~$ is, for example, $z=x$. So for the statement to be incorrect we need $\\neg(x\\le 2x)$, that is, $x>2x$, which (by subtracting $x$ on both sides) holds if and only if $0>x$. So any negative integer $x$ makes Statement 2 false and is therefore a counterexample to Statement 1.", "Statement 1 is indeed incorrect. If $x\\le 2x$, then a suitable $z$ so that $x\\le z\\le 2x~$ is, for example, $z=x$. So for the statement to be incorrect we need $\\neg(x\\le 2x)$, that is, $x>2x$, which (by subtracting $x$ on both sides) holds if and only if $0>x$. So any negative integer $x$ makes Statement 2 false and is therefore a counterexample to Statement 1."], "matrix": ["0", "1"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "shuffleChoices": false, "scripts": {}, "type": "1_n_2", "prompt": "In view of your answer to c), is Statement 1 in a) above true?
"}, {"choices": ["For every integer $x$ there exists an integer $z$ such that $z<x$ or $z>2x$.
", "For every integer $z$ there exists an integer $x$ such that $z<x$ or $z>2x$.
", "There exists an integer $x$ such that there is an integer $z$ so that $z<x$ or $z>2x$.
", "There exists an integer $x$ such that for all integers $z$ we have $z<x$ or $z>2x$.
", "There exists an integer $x$ such that for all integers $z$ we have $x>z>2x$.
"], "displayColumns": 0, "minMarks": 0, "showFeedbackIcon": true, "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "maxMarks": 0, "distractors": ["Incorrect. The order of the quantifiers $\\forall x~\\exists z$ matters.", "Incorrect. The names of the variables in $\\forall x~\\exists z$ matter.", "Incorrect: $\\exists x~\\exists z~$ cannot be the negation of $\\forall x~\\exists z~$.", "", "Incorrect: $x\\le z\\le 2x$ stands for \"$x\\le z$ and $z\\le 2x$\" whose negation is \"$x>z$ or $z>2x$\"."], "matrix": ["-1", "-1", "-1", "1", "-1"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "shuffleChoices": false, "scripts": {}, "type": "1_n_2", "prompt": "We consider the negation of Statement 1 in a).
\nWhich of the following is a correct formulation of this negated statement? Choose any that apply.
"}], "extensions": [], "functions": {}, "variables": {}, "ungrouped_variables": [], "statement": "This question is about statements with quantifiers $\\forall$ and $\\exists$ and their negations.
The set of all integers is denoted by $\\mathbb Z$.
Click on \"Show feedback\" after \"Submit part\" for detailed explanations.
", "tags": [], "metadata": {"description": "Example of a universal statement over the integers and its negation
", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "rulesets": {"": []}, "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"js": "", "css": ""}, "name": "Daniel's copy of statements with quantifiers and their negations", "type": "question", "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}]}]}], "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}]}