\\[S = \\left\\{\\simplify[std]{({a}n^2+{a1})/({b}n^2+{b1})}\\;\\;:\\;\\;n \\in \\mathbb{Z} \\right\\}\\]
\nGreatest lower bound = [[0]] (Enter as a fraction or integer, not a decimal.)
\nLeast upper bound = [[1]] (Enter as a fraction or integer, not a decimal.)
\nDoes the glb lie in the set? [[2]]
\nDoes the lub lie in the set? [[3]]
\n\n
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\nGreatest lower bound = [[0]]
\nDoes this lie in the set? [[1]]
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\nDoes this lie in the set? [[3]]
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\nDoes this lie in the set? [[1]]
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\nDoes this lie in the set? [[3]]
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\n\n\n\nGreatest lower bound = [[0]]
\n\n\n\nDoes this lie in the set? [[1]]
\n\n\n\nLeast upper bound = [[2]] $\\;\\;\\;\\;$
\n\n\n\nDoes this lie in the set? [[3]]
\n\n", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{ans4}", "minValue": "{ans4}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"displayType": "radiogroup", "choices": ["Yes
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\nIf the set is not bounded above, enter the lub as infinity by typing in the word infinity.
If the set is not bounded below, enter the glb as -infinity by typing in -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 the set.
\nThere are $8$ 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", "MAS1601", "mas1601", "max value", "maximum value", "min value", "minimum value", "not bounded", "query", "sets", "tested1", "upper bound"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "4/07/2012:
\nAdded tags. Corrected tags.
\nCorrected mistake in answer to first part (minus sign missing).
\n5/07/2012:
\nThere is an issue with the MCQs - this has been reported on Github.
\nAlso an issue with recognising infinity as an answer - also reported on Github.
\nChanged to Match Text Pattern, but Correct Answer not properly displayed for $\\pm \\infty$
\nAlso an issue with reordering gaps in a gapfill - wishlist item on Github
\nAdvice display tidied up.
\n21/07/2012:
\nError in part c first MCQ. Corrected.
\nInstructions about using fractions and integers included.
\nAdded description.
\nHave used Matching Expressions question typefor identifying $\\pm \\infty$ as answers.
\n27/7/2012:
\nAdded tags.
\nEdited grammar in Advice section.
\n24/12/2012:
\nChecked calculations. Added tested1 tag.
\nQuestion 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.
\nTested rounding, OK. Added cr1 tag.
\nC
", "licence": "Creative Commons Attribution 4.0 International", "description": "Eight questions on finding least upper bounds and greatest lower bounds of various sets.
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\\[\\begin{eqnarray*} \\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{eqnarray*} \\]
Note that 1) the values for positive and negative values of $n$ are the same and 2) as $n$ increases this expression increases.
The 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 $\\simplify[std]{{a}/{b}}$.
\nb)
\\[\\begin{eqnarray*} \\simplify[std]{{c} * x ^ {2 * m + 1}}&\\lt&\\simplify[std]{ {d} * x ^ {2 * m}} \\Leftrightarrow\\\\ \\simplify[std]{x ^ {2 * m} * ({c} * x -{d})} &\\lt& 0 \\Leftrightarrow\\\\ \\simplify[std]{{c}x-{d}} &\\lt& 0 \\textrm{ as }x^{\\var{2*m}} \\geq 0 \\end{eqnarray*} \\]
Hence this set is the same as the set
\\[\\left \\{x \\in \\mathbb{R}\\;\\;:\\;\\;x \\lt \\simplify[std]{{d}/{c}}\\right\\}\\]
This set does not have a greatest lower bound so you enter -infinity.
It does have a least upper bound and this is $\\simplify[std]{{d}/{c}}$
\nc)
\\[S = \\left\\{\\simplify[std]{{a2}+{s1}*{b2}/n^{r}}\\;\\;:\\;\\;n \\in \\mathbb{N} \\right\\}\\]
Let $\\displaystyle a_n=\\simplify[std]{{a2}+{s1}*{b2}/n^{r}}$
As $n$ increases we see that $a_n$ {mo}creases and converges to the limit $\\var{a2}$.
\nHence greatest lower bound = $\\var{glb3}$ and least upper bound = $\\var{lub3}$
\nd)
\\[S = \\left\\{\\simplify[std]{{a4}x+{b4}/x}\\;\\;:\\;\\;x \\in \\mathbb{R},\\;\\;x \\gt 0 \\right\\}\\]
It is clear that this set does not have a least upper bound, so we enter infinity for this value.
However it does have a lower bound as we have $\\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 as $x \\gt 0$).
\nIt is not hard to see that this gives a minimum value for $g(x)$ and $g(\\var{r5})=\\var{ans4}$.
\nHence the greatest lower bound is $\\var{ans4}$ as $g(x) \\geq \\var{ans4},\\;\\;\\forall x \\gt 0$.
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Questions on the least upper bounds and greatest lower bounds of sets of the form $\\{ f(x) : x \\in \\mathbb{Z} \\text{ or } \\mathbb{R} \\}$.
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