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Questions on using quantifiers.

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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 $0$.

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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]}

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", "

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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]}

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", "

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]}

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"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"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).

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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.

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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.

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

", "

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

", "

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

", "

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

"], "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

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

", "

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

", "

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

", "

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

"], "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: $\\displaystyle \\exists a \\in \\mathbb{R^+},\\;\\exists b \\in \\mathbb{N},\\;\\exists c \\in \\mathbb{N}\\;\\left[(c \\lt b+1) \\land \\left(\\frac{1}{2^n} \\geq 3a\\right)\\right]$

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\\\\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 $0$.

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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.

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