Find least upper bounds and greatest lower bounds of various sets

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Question statement

For each of the following sets $S$ , state the least upper bound (lub) and the greatest lower bound (glb), where appropriate.

If 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.

Also state if the lub or glb belong to the set.

There are $8$ parts to this question, so you may need to scroll down to answer all parts.

Give any introductory information the student needs.

			Name	Type	Generated Value

ans4	integer	90
b4	number	225
b5	integer	-9
b6	integer	-1
b7	integer	4
b1	integer	7
b2	integer	7
b8	integer	7
s9	integer	1
s2	integer	1
s1	integer	-1
glb7	number	2.35
glb6	number	2.95
glb3	integer	0
a1	integer	-7
a2	integer	7
a5	integer	1
a4	integer	9
a7	integer	20
a6	integer	5
a8	integer	9
c6	integer	3
a	integer	8
r5	integer	5
c	integer	1
b	integer	3
lub3	integer	7
d	integer	-2
mo	string	in
m	integer	4
s	integer	-1
r	integer	6
perhaps1	string	Yes
perhaps2	string	No
sg	integer	0
something	string	positive

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
2*a4*r5

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `integer`

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Parts

Part a) Gap-fill
Part b) Gap-fill
Part c) Gap-fill
Part d) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Choose one from a list
  Add
3. Gap Gap 2. Mathematical expression
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4. Gap Gap 3. Choose one from a list
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

$S = \left\{\simplify[std]{({a}n^2+{a1})/({b}n^2+{b1})}\;\;:\;\;n \in \mathbb{Z} \right\}$

Greatest lower bound = Gap 0. (Enter as a fraction or integer, not a decimal.)

Least upper bound = Gap 1. (Enter as a fraction or integer, not a decimal.)

Does the glb lie in the set? Gap 2.

Does the lub lie in the set? Gap 3.

Advice

$\begin{eqnarray*} \simplify[std]{({a}n^2+{a1})/({b}n^2+{b1})}&=& \simplify[std]{(({a} / {b}) * ({b} * n ^ 2 + {b1}) + {a1} -({a * b1} / {b})) / ({b} * n ^ 2 + {b1})}\\ &=& \simplify[std]{{a} / {b} -({( -a1) * b + a * b1} / ({b} * ({b} * n ^ 2 + {b1})))}\\ \end{eqnarray*}$
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 $\displaystyle \simplify[std]{{a1}/{b1}}$ .

As $n$ increases, the value of the expression approaches as close as we like to $\displaystyle \simplify[std]{{a}/{b}}$ , but is always less than $\displaystyle \simplify[std]{{a}/{b}}$ .

Hence the least upper bound is $\simplify[std]{{a}/{b}}$ .

$\begin{eqnarray*} \simplify[std]{{c} * x ^ {2 * m + 1}}&\lt&\simplify[std]{ {d} * x ^ {2 * m}} \Leftrightarrow\\ \simplify[std]{x ^ {2 * m} * ({c} * x -{d})} &\lt& 0 \Leftrightarrow\\ \simplify[std]{{c}x-{d}} &\lt& 0 \textrm{ as }x^{\var{2*m}} \geq 0 \end{eqnarray*}$

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 $\simplify[std]{{d}/{c}}$

$S = \left\{\simplify[std]{{a2}+{s1}*{b2}/n^{r}}\;\;:\;\;n \in \mathbb{N} \right\}$
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 $\var{a2}$ .

Hence greatest lower bound = $\var{glb3}$ and least upper bound = $\var{lub3}$

$S = \left\{\simplify[std]{{a4}x+{b4}/x}\;\;:\;\;x \in \mathbb{R},\;\;x \gt 0 \right\}$

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 $\displaystyle \var{a4}x+\frac{\var{b4}}{x} \gt 0,\;\;\forall x \gt 0$ .

To find the greatest lower bound we find the minimum value of $\displaystyle g(x)=\var{a4}x+\frac{\var{b4}}{x},\;\;x \gt 0$ .

Now $\displaystyle g'(x)=\var{a4}-\frac{\var{b4}}{x^2}$ and $g'(x)=0$ when $\displaystyle x=\sqrt{\frac{\var{b4}}{\var{a4}}} = \var{r5}$ .

(We take the positive square root as $x \gt 0$ ).

It is not hard to see that this gives a minimum value for $g(x)$ and $g(\var{r5})=\var{ans4}$ .

Hence the greatest lower bound is $\var{ans4}$ as $g(x) \geq \var{ans4},\;\;\forall x \gt 0$ .

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This question is used in the following exam:

Bounds of sets by Newcastle University Mathematics and Statistics in Content created by Newcastle University.

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Find least upper bounds and greatest lower bounds of various sets

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Generated value: `integer`

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Find least upper bounds and greatest lower bounds of various sets

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Generated value: integer

Parts

Gap-fill

Generated value: `integer`