Find least upper bounds and greatest lower bounds of four sets

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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.

Enter each number as a fraction or integer, not a decimal.

If 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.

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

There are four 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	30
b4	number	45
b5	integer	5
b6	integer	1
b7	integer	8
b1	integer	4
b2	integer	9
b8	integer	15
s9	integer	-1
s2	integer	1
s1	integer	1
glb7	number	0.74
glb6	number	3.9166666667
glb3	integer	6
a1	integer	-4
a2	integer	6
a5	integer	8
a4	integer	5
a7	integer	13
a6	integer	3
a8	integer	20
c6	integer	4
a	integer	8
r5	integer	3
c	integer	2
b	integer	5
lub3	integer	15
d	integer	3
m	integer	4
s	integer	1
r	integer	4
perhaps1	string	No
perhaps2	string	Yes
sg	integer	0
something	string	negative
part_c_glb_marking	list	[ 0, 1 ]
part_c_lub_marking	list	[ 1, 0 ]

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
2*a4*r5

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Generated value: `integer`

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Gap-fill

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

Greatest lower bound = Gap 0.

Does this lie in the set? Gap 1.

Least upper bound = Gap 2.

Does this lie in the set? Gap 3.

Advice

a)

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

Note that 1) the values for positive and negative values of $n$ are the same and 2) as $n$ increases this expression increases.

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 $\simplify[std]{{a}/{b}}$ .

b)

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

Hence this set is the same as the set

$\left \{x \in \mathbb{R} \; : \; x \lt \simplify[std]{{d}/{c}}\right\}$

This set does not have a greatest lower bound so you enter -infinity.

It does have a least upper bound and this is $\simplify[std]{{d}/{c}}$

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 $a_n$ {if(s1=1,'decreases','increases')} and converges to the limit $\var{a2}$ .

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

d)

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

This set does not have a least upper bound, so you enter infinity.

However, it does have a lower bound because $\displaystyle \var{a4}x+\frac{\var{b4}}{x} \gt 0$ , $\forall x \gt 0$ .

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, since $x \gt 0$ )

This gives a minimum value for $g(x)$ at $x = \var{r5}$ , and $g(\var{r5})=\var{ans4}$ .

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

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Bill Foster 6 years, 4 months ago

Newcastle University Mathematics and Statistics 9 years, 3 months ago

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Find least upper bounds and greatest lower bounds of four sets

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Taxonomy: mathcentre

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Bill Foster 6 years, 4 months ago

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

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Generated value: integer

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Gap-fill

a)

b)

c)

d)

Generated value: `integer`