This question is about statements with quantifiers $\\forall$ and $\\exists$ and their negations.
The set of all integers is denoted by $\\mathbb Z$.
$\\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 $
"], "maxAnswers": "4", "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.
"}, {"matrix": ["2", 0], "showCorrectAnswer": true, "shuffleChoices": false, "variableReplacements": [], "marks": 0, "minMarks": 0, "scripts": {}, "displayType": "radiogroup", "maxMarks": "2", "displayColumns": 0, "distractors": ["For $x=0$ and $z=0$ the statement is true.", "For $x=0$ and $z=0$ the statement is true."], "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "choices": ["yes
", "no
"], "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 $x=0$?
"}, {"minValue": "-19999999999999999", "correctAnswerFraction": false, "showCorrectAnswer": true, "integerAnswer": true, "type": "numberentry", "variableReplacements": [], "scripts": {}, "integerPartialCredit": 0, "allowFractions": false, "marks": "3", "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):
", "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false}, {"matrix": ["0", "3"], "showCorrectAnswer": true, "shuffleChoices": false, "variableReplacements": [], "marks": 0, "minMarks": 0, "scripts": {}, "displayType": "radiogroup", "maxMarks": "3", "displayColumns": 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."], "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "choices": ["yes
", "no
"], "prompt": "In view of your answer to c), is Statement 1 in a) above true?
"}, {"matrix": ["-4", "-4", "-4", "6", "-3"], "showCorrectAnswer": true, "shuffleChoices": true, "variableReplacements": [], "marks": 0, "minMarks": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst", "warningType": "none", "displayType": "checkbox", "maxMarks": "6", "displayColumns": "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: $\\exists x~\\exists z~$ 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 $z>2x$\"."], "minAnswers": 0, "type": "m_n_2", "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 $x>z>2x$.
\n"], "maxAnswers": "5", "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.
"}, {"matrix": ["-4", "6", "-4", "-4", "-4"], "showCorrectAnswer": true, "shuffleChoices": true, "variableReplacements": [], "marks": 0, "minMarks": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst", "warningType": "none", "displayType": "checkbox", "maxMarks": "6", "displayColumns": "1", "distractors": ["Incorrect, because $x\\le z\\le 2x$ stands for $(x\\le z)\\wedge (z\\le 2x)$ whose negation is $\\neg(x\\le z)\\vee \\neg(z\\le 2x)$.", "Correct! Note that $x\\le z\\le 2x$ stands for $(x\\le z)\\wedge (z\\le 2x)$ whose negation is $\\neg(x\\le z)\\vee \\neg(z\\le 2x)$.", "Incorrect. The negation of $\\exists z~P(z)$ is $\\forall z~\\neg P(z)$.", "Incorrect. The negation of $\\forall x~\\exists z~Q(x,z)~$ is $~\\exists x~\\forall z~\\neg Q(x,z)$.", "There are two mistakes here. The quantified variables $x$ and $z$ must appear in their original order, and $x\\le z\\le 2x$ stands for $(x\\le z)\\wedge (z\\le 2x)$ whose negation is $\\neg(x\\le z)\\vee \\neg(z\\le 2x)$."], "minAnswers": 0, "type": "m_n_2", "choices": ["$\\exists x\\in \\mathbb Z~:~\\forall z\\in \\mathbb Z~:~(z<x)\\wedge (z>2 x)$
", "$\\exists x\\in \\mathbb Z~:~\\forall z\\in \\mathbb Z~:~(z<x)\\vee (z>2 x)$
", "$\\exists x\\in \\mathbb Z~:~\\exists z\\in \\mathbb Z~:~(z<x)\\vee (z>2 x)$
", "$\\forall x\\in \\mathbb Z~:~\\exists z\\in \\mathbb Z~:~(z<x)\\vee (z>2 x)$
", "$\\exists z\\in \\mathbb Z~:~\\forall x\\in \\mathbb Z~:~x>z>2 x$
"], "maxAnswers": "6", "prompt": "Finally, find the correct formulation of the negation of Statement 1 in terms of mathematical symbols. Choose any that apply.
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