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Enter as a fraction or integer, not as a decimal.
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\nGreatest lower bound = [[0]]
\nDoes this lie in the set? [[1]]
\nLeast upper bound = [[2]]
\nDoes this lie in the set? [[3]]
\n", "marks": 0}, {"scripts": {}, "gaps": [{"answer": "-infinity", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"displayType": "radiogroup", "choices": ["Yes
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\nGreatest lower bound = [[0]]
\nDoes this lie in the set? [[1]]
\nLeast upper bound = [[2]]
\nDoes this lie in the set? [[3]]
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\nGreatest lower bound = [[0]]
\nDoes this lie in the set? [[1]]
\nLeast upper bound = [[2]]
\nDoes this lie in the set? [[3]]
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\nGreatest lower bound = [[0]]
\nDoes this lie in the set? [[1]]
\nLeast upper bound = [[2]]
\nDoes this lie in the set? [[3]]
", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "For each of the following sets $S$ , state the least upper bound (lub) and the greatest lower bound (glb), where appropriate.
\nEnter each number as a fraction or integer, not a decimal.
\nIf the set is not bounded above, write infinity
.
If the set is not bounded below, write -infinity
.
$\\mathbb{N}$ denotes the set of natural numbers, $\\mathbb{Z}$ the set of integers and $\\mathbb{R}$ the set of real numbers.
\nAlso state if the lub or glb belong to each set.
\nThere are four parts to this question, so you may need to scroll down to answer all parts.
", "tags": ["bounded above", "bounded below", "bounded set", "bounds", "checked2015", "cr1", "glb", "greatest lower bound", "least upper bound", "limit", "limits", "lower bound", "lub", "MAS2224", "max value", "maximum value", "min value", "minimum value", "not bounded", "query", "sets", "tested1", "upper bound"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t4/07/2012:
\n\t\tAdded tags. Corrected tags.
\n\t\tCorrected mistake in answer to first part (minus sign missing).
\n\t\t5/07/2012:
\n\t\tThere is an issue with the MCQs - this has been reported on Github.
\n\t\tAlso an issue with recognising infinity as an answer - also reported on Github.
\n\t\tChanged to Match Text Pattern, but Correct Answer not properly displayed for $\\pm \\infty$
\n\t\tAlso an issue with reordering gaps in a gapfill - wishlist item on Github
\n\t\tAdvice display tidied up.
\n\t\t21/07/2012:
\n\t\tError in part c first MCQ. Corrected.
\n\t\tInstructions about using fractions and integers included.
\n\t\tAdded description.
\n\t\tHave used Matching Expressions question typefor identifying $\\pm \\infty$ as answers.
\n\t\t27/7/2012:
\n\t\tAdded tags.
\n\t\tEdited grammar in Advice section.
\n\t\t24/12/2012:
\n\t\tChecked calculations. Added tested1 tag.
\n\t\tQuestion now accepts infinity and -infinity as possible answers. Query raised as could use html code for infinity to display correct answer for the Word Match rather than the input strings. However, these are not the same as the student would input. Added query tag.
\n\t\tTested rounding, OK. Added cr1 tag.
\n\t\tC
\n\t\t20/01/2014:
\n\t\tGot rid of last four parts.
\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Four questions on finding least upper bounds and greatest lower bounds of various sets.
"}, "advice": "\\begin{align}
\\simplify[std]{({a}n^2+{a1})/({b}n^2+{b1})} &= \\simplify[std]{(({a} / {b}) * ({b} * n ^ 2 + {b1}) + {a1} -({a * b1} / {b})) / ({b} * n ^ 2 + {b1})} \\\\
&= \\simplify[std]{{a} / {b} -({( -a1) * b + a * b1} / ({b} * ({b} * n ^ 2 + {b1})))}
\\end{align}
Note that 1) the values for positive and negative values of $n$ are the same and 2) as $n$ increases this expression increases.
\nThe greatest lower bound occurs when $n=0$ and the value is $\\displaystyle \\simplify[std]{{a1}/{b1}}$.
\nAs $n$ increases, the value of the expression approaches as close as we like to $\\displaystyle \\simplify[std]{{a}/{b}}$ , but is always less than $\\displaystyle \\simplify[std]{{a}/{b}}$.
\nHence the least upper bound is $\\simplify[std]{{a}/{b}}$.
\n\\begin{align}
&&\\simplify[std]{{c} * x ^ {2 * m + 1}} &\\lt \\simplify[std]{ {d} * x ^ {2 * m}} \\\\
\\iff && \\simplify[std]{x ^ {2 * m} * ({c} * x -{d})} &\\lt 0 \\\\
\\iff &&\\simplify[std]{{c}x-{d}} &\\lt 0 \\text{ as } x^{\\var{2*m}} \\geq 0
\\end{align}
Hence this set is the same as the set
\n\\[\\left \\{x \\in \\mathbb{R} \\; : \\; x \\lt \\simplify[std]{{d}/{c}}\\right\\}\\]
\nThis set does not have a greatest lower bound so you enter -infinity
.
It does have a least upper bound and this is $\\simplify[std]{{d}/{c}}$
\n\\[S = \\left\\{\\simplify[std]{{a2}+{s1}*{b2}/n^{r}} \\; : \\; n \\in \\mathbb{N} \\right\\}\\]
\nLet $\\displaystyle a_n=\\simplify[std]{{a2}+{s1}*{b2}/n^{r}}$
\nAs $n$ increases we see that $a_n$ {if(s1=1,'decreases','increases')} and converges to the limit $\\var{a2}$.
\nHence, greatest lower bound = $\\var{glb3}$ and least upper bound = $\\var{lub3}$
\n\\[S = \\left\\{\\simplify[std]{{a4}x+{b4}/x} \\; : \\; x \\in \\mathbb{R}, \\; x \\gt 0 \\right\\}\\]
\nThis set does not have a least upper bound, so you enter infinity
.
However, it does have a lower bound because $\\displaystyle \\var{a4}x+\\frac{\\var{b4}}{x} \\gt 0$, $\\forall x \\gt 0 $.
\nTo find the greatest lower bound we find the minimum value of $\\displaystyle g(x)=\\var{a4}x+\\frac{\\var{b4}}{x}$, $x \\gt 0$.
\nNow $\\displaystyle g'(x)=\\var{a4}-\\frac{\\var{b4}}{x^2}$ and $g'(x)=0$ when $\\displaystyle x=\\sqrt{\\frac{\\var{b4}}{\\var{a4}}} = \\var{r5}$.
\n(We take the positive square root, since $x \\gt 0$)
\nThis gives a minimum value for $g(x)$ at $x = \\var{r5}$, and $g(\\var{r5})=\\var{ans4}$.
\nHence the greatest lower bound is $\\var{ans4}$ as we have shown that $g(x) \\geq \\var{ans4}$, $\\forall x \\gt 0$.
", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}