// Numbas version: exam_results_page_options {"name": "Numbastest uke 33: Mengder", "metadata": {"description": "", "licence": "None specified"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], [], [], [], [], []], "questions": [{"name": "3 - Elementer og mengder", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

La $A=\\{\\var{a},\\var{b},\\var{c}\\}$ og $B=\\{\\var{a},\\var{c}\\}$.

", "advice": "

Svarene står ovenfor, men la oss raskt forklare symbolene. $\\in$ betyr ‹element›, og $\\var{a}\\in B$ spør enkelt og greit om $\\var{a}$ er et element i $B$. Med andre ord om $\\var{a}$ er inneholdt i $B$. Symbolet $\\subseteq$ minner om krokodillemunnen fra barneskolen, og betyr delmengde eller undermengde. $B\\subseteq A$ spør om alle elementene i $A$ er inneholdt i $B$,dersom dette er sant er $A$ en undermengde av $B$.

Hvilke utsagn er sanne?

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La $R=\\{\\var{a},\\var{b},\\var{c},\\var{d}\\}$ og $S=\\{\\var{b},\\var{c},\\var{d},\\var{f}\\}$.

", "advice": "

Oppgave a) $R\\cap S$: Her leses symbolet $\\cap$ som snitt, det er altså snakk bare det som ligger i begge mengdene. Dersom noe ligger i bare en av mengdene skal det altså ikke tas med.

$R=\\{\\var{a},\\var{b},\\var{c},\\var{d}\\}$

$S=\\{\\var{b},\\var{c},\\var{d},\\var{f}\\}$

$R\\cap S=\\{\\var{b},\\var{c}\\}$

Oppgave b) $R\\cup S$:Husk at du kan tenke på $\\cup$ som addisjon. Vi ønsker å legge sammen alle elementene i $R$ og $S$. NB:husk at i en mengde er alle elementene unike så vi skriver ikke ett element mer enn èn gang.

$R=\\{\\var{a},\\var{b},\\var{c},\\var{d}\\}$

$S=\\{\\var{b},\\var{c},\\var{d},\\var{f}\\}$

$R\\cup S=\\{\\var{a},\\var{b},\\var{c},\\var{d},\\var{f}\\}$

Oppgave c) $R\\setminus S$: Her leses $\\setminus$ som komplimentet til. Dette kan en litt feilaktig tenke på som subtraksjon eller minus. Vi ønsker med andre ord alt som ligger i $R$ men ikke i $S$:

$R=\\{\\var{a},\\var{b},\\var{c},\\var{d}\\}$

$S=\\{\\var{b},\\var{c},\\var{d},\\var{f}\\}$

$R\\setminus S=\\{\\var{a}\\}$

Oppgave d) $S\\setminus R$: Her leses $\\setminus$ som komplimentet til. Dette kan en litt feilaktig tenke på som subtraksjon eller minus. Vi ønsker med andre ord alt som ligger i $S$ men ikke i $R$:

$R=\\{\\var{a},\\var{b},\\var{c},\\var{d}\\}$

$S=\\{\\var{b},\\var{c},\\var{d},\\var{f}\\}$

$S\\setminus R=\\{\\var{f}\\}$

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Finn $R\\cap S=\\{$[[0]],[[1]],[[2]]$\\}$

Finn $R\\cup S=\\{$[[0]],[[1]],[[2]],[[3]],[[4]]$\\}$

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Finn $R \\setminus S=\\{$[[0]]$\\}$

Finn $S\\setminus R=\\{$[[0]]$\\}$

Gitt mengden $M=\\{\\var{a},\\var{a+1},\\var{a+2},\\var{a+3},\\var{a+4},\\var{a+5},\\var{a+6}\\}$.

Skriv følgende mengder på listeform:

", "advice": "

Oppgave a) Ønsker å skrive mengden $A=\\{x\\in M| x\\neq\\var{a+1}, x\\neq\\var{a+4}\\}$ på listeform. Notasjonen kan og leses slik: Ønsker alle elementer i $M$ slik at elementeneer ulik $\\var{a+1}$og $\\var{a+4}$. Hvor tegnet $|$ leses som ‹slik at›. Dette er selvsagt mengden $A=\\{\\var{a},\\var{a+2},\\var{a+3},\\var{a+5},\\var{a+6}\\}$.

Oppgave b) Ønsker å skrive mengden $B=\\{x+7|x\\in M\\}$ på listeform. Her kan vi tenke oss at vi ønsker å gjøre $x+7$ gitt at $x$ ligger i $M$. Altså må vi taalle verdiene i mengden $M$ og legge til 7: $B=\\{\\var{a+7},\\var{a+8},\\var{a+9},\\var{a+10},\\var{a+11},\\var{a+12},\\var{a+13}\\}$.

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$A=\\{x\\in M| x\\neq\\var{a+1}, x\\neq\\var{a+4}\\}$

$A=\\{$[[0]],[[1]],[[2]],[[3]],[[4]]$\\}$

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$B=\\{x+7|x\\in M\\}$

$B=\\{$[[0]],[[1]],[[2]],[[3]],[[4]],[[5]],[[6]]$\\}$

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Simple questions on interval notation. If you are not randomising the order of your questions please turn on randomise choices in these questions.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Hvilke av følgende beskriver intervallet $(\\var{a},\\var{b})$?

", "advice": "

Åpen (runde)

De åpne (runde) parentesene i $(\\var{a},\\var{b})$ betyr at disse endepunktene ikke er inkludert i intervallet (men alle reelle tall mellom dem er inkludert). Dette er kjent som det åpne intervallet $(\\var{a},\\var{b})$.

delta

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Alle reelle tall mellom $\\var{a}$ og $\\var{b}$ inkludert $\\var{a}$ og $\\var{b}$.

", "

Alle reelle tall mellom $\\var{a}$ og $\\var{b}$ ekskludert $\\var{a}$ og $\\var{b}$.

", "

Alle reelle tall mellom $\\var{a}$ og $\\var{b}$ med $\\var{a}$ men uten $\\var{b}$.

", "

Alle reelle tall mellom $\\var{a}$ og $\\var{b}$ med $\\var{b}$ men uten $\\var{a}$.

"], "matrix": ["0", "1", 0, 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "5 - Match mengde og intervall", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

Hvilke av mengder er like?

", "advice": "", "rulesets": {}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-2 .. 1#1)", "description": "", "templateType": "randrange"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2 .. 5#1)", "description": "", "templateType": "randrange"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_x", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": false, "shuffleAnswers": false, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["$A=\\{x\\in\\mathbb{R}|\\var{a}\\leq x<\\var{b}\\}$", "$B=\\{x\\in\\mathbb{R}|\\var{a}< x\\leq\\var{b}\\}$", "$C=\\{x\\in\\mathbb{R}| x\\leq\\var{b}\\}$", "$D=\\{x\\in\\mathbb{R}|\\var{a}< x\\}$"], "matrix": [[0, "1", 0, 0], [0, 0, "1", 0], [0, 0, 0, "1"], ["1", 0, 0, 0]], "layout": {"type": "all", "expression": ""}, "answers": ["$E=(\\var{a},\\infty)$", "$F=[\\var{a},\\var{b})$", "$G=(\\var{a},\\var{b}]$", "$H=(-\\infty,\\var{b}]$"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "4 - Intervaller og paranteser ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

Skriv følgende mengder som intervaller: (bruk -infty for å skrive inn $-\\infty$ og infty for å skrive inn $\\infty$)

", "advice": "

1) $\\{x\\in\\mathbb{R}|x\\leq\\var{b}\\}$ . Her ønsker vi enkelt og greit alle reelle tall mindre eller lik $\\var{b}$: $x\\in(-\\infty,\\var{b}]$, hvor ( ble brukt for å ekskludere $-\\infty$ mens ] ble brukt for å inkludere $\\var{b}$.

2) Mengden $\\{x\\in\\mathbb{R}|\\var{a}\\leq x\\leq\\var{b}\\}$ kan leses som at vi ønsker alle relle tall som er større eller lik $\\var{a}$ og mindre eller lik $\\var{b}$: $x\\in[\\var{a},\\var{b}]$, hvor [ ble brukt for å vise at dette er ett lukket intervall; altså ett intervall der endepunktet er inkludert.

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sett inn

( eller [

nedre

grense

øvre

grense

sett inn

) eller ]

$\\{x\\in\\mathbb{R}|x\\leq\\var{b}\\}$

[[0]]

[[1]]

[[2]]

[[3]]

$\\{x\\in\\mathbb{R}|\\var{a}\\leq x\\leq\\var{b}\\}$

[[4]]

[[5]]

[[6]]

[[7]]

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Snitt og union av intervaller", "extensions": [], "custom_part_types": [], "resources": [["question-resources/intervaller.PNG", "/srv/numbas/media/question-resources/intervaller.PNG"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ida Friestad Pedersen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/792/"}, {"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

I den loddrette kolonnen er det gitt union eller snitt av noen ulike mengder (hovedsakelig intervaller). I den vannrette linjen er det oppgitt enklere mengder (i de fleste tilfeller uttrykt som intervaller).

For hver linje i den loddrette listen skal dere velge hvilket resultat dere mener er riktig.

", "advice": "

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Velg notasjoner for følgende tallmengdene:

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Velg en riktig operatør eller forbindelse for hver påstand

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La $a$, $b$, $x$ og $y$ stå for vilkårlige reelle tall.

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Avgjør hvilke av implikasjonene som gjelder

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$x>b-a$
", "$a^2+b^2=0$

$a=0\\ \\ \\mathrm{og}\\ \\ b=0$", "$a^2+b^2=0$

$a=0\\ \\ \\mathrm{eller}\\ \\ b=0$", "$x^2>0$

$x>0$ ", "$x=2$

$x^4=16$ ", "$x^4=16$

$x=2$ eller $x=-2$", "$x=-2$ eller $x=2$

$x\\in\\{-2,2\\}$", "$x\\in\\{-2,2\\}$

$x\\in\\{-2,2,3\\}$", "$x=-2$ eller $x=2$

$x=-2$ eller $x=2$ eller $x=3$", "$x^4=16$

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