// Numbas version: exam_results_page_options {"name": "Quantifiers 3-", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Quantifiers 3-", "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.

", "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.

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