Daniel's copy of Quantifiers1-

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Kimia Aksir

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4 years, 7 months ago

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Kimia Aksir 4 years, 7 months ago

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Daniel Mansfield 7 years, 7 months ago

Created this as a copy of Quantifiers1-.

Name	Status	Author	Last Modified
Quantifiers1-	Ready to use	Newcastle University Mathematics and Statistics	20/11/2019 14:51
Daniel's copy of Quantifiers1-	Ready to use	Daniel Mansfield	01/09/2020 01:50
Luis's copy of Quantifiers1-	draft	Luis Hernandez	30/11/2018 18:13
Luis's copy of Quantifiers1-	draft	Luis Hernandez	30/11/2018 18:17

There are 6 other versions that do you not have access to.

Question statement

Choose the appropriate proposition for the following English sentences. Also choose whether they are true or false.

You must choose two of the five options presented in each row. You may need to scroll over to see all five options.

Note also that every wrong answer takes away from your score. However, your minimum score is $0$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

			marking_matrix	list	Nested 3×5 list
			select	list	[ 3, 8, 7 ]

			Name	Type	Generated Value

			select1	list	[ 0, 1, 2 ]
			marking_matrix1	list	Nested 3×5 list

			Name	Type	Generated Value

			select2	list	[ 4, 5, 6 ]
			marking_matrix2	list	Nested 3×5 list

			Name	Type	Generated Value

			all	list	Nested 9×3 list
			neg_marks	matrix	Matrix of size 3×3

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
map(list(neg_marks[x])+[all[select[x]][2]]+[-1*all[select[x]][2]],x,0..2)

JME function reference

Description

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

Can an exam override the value of this variable?

Generated value: `list`

[ [ 0.5, -0.5, -0.5, -0.5, 0.5 ], [ -0.5, 0.5, -0.5, -0.5, 0.5 ], [ -0.5, -0.5, 0.5, 0.5, -0.5 ] ]

← Depends on:

all
neg_marks
select

→ Used by:

"Part a)" - Custom matrix expression

Match choices with answers

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

Minimum marks

Maximum marks

(no maximum)

Show correct answer on reveal?

Show score feedback icon?

Advanced settings

Minimum answers

Maximum answers

What to do if wrong number of answers selected:

Selection type

Layout

Show choice feedback state?

{all[select[0]][0]}
{all[select[1]][0]}
{all[select[2]][0]}

{all[select[0]][1]}
{all[select[1]][1]}
{all[select[2]][1]}
True
False

For each combination of answer and choice, specify the number of marks to add or subtract when the student picks it.

	{all[select[0]][1]}	{all[select[1]][1]}	{all[select[2]][1]}	True	False
{all[select[0]][0]}
{all[select[1]][0]}
{all[select[2]][0]}

Advice

$\forall x \in \mathbb{R}\;(x^2\gt 0)$ is false, because there is a single counterexample when $x=0$ (in which case $x^2=0$ , which is not allowed in the strict inequality).

$\forall x \in \mathbb{R} \;\exists n \in \mathbb{N}\;(x^n\geq 0)$ is true, because it is always true for any even value of $n$ . As seen in the previous example, $n=2$ will work here because the inequality is no longer strict. You might also consider the simpler case $n=0$ , though $x^0$ is not well-defined when $x=0$ .

$\forall X \subseteq \mathbb{N}\;(X\subseteq \mathbb{R})$ is true, because $\mathbb{N} \subset \mathbb{R}$ itself - that is, all natural numbers are also real numbers.

$\forall n \in \mathbb{N}\;\exists X \subseteq \mathbb{N}\;(|X|\lt n)$ is false, because there is a single counterexample when $n=0$ (in which case $|X|$ cannot possibly be negative). Notice $n=1$ is not a counterexample since $|\emptyset| = 0 < 1$ .

$\exists n \in \mathbb{N}\; \forall X \subseteq \mathbb{N}\;(|X|\lt n)$ is false, because for any given natural number $n$ , we could take $X$ to be the set $\{0,1,2,\ldots,n\}$ , which has size $n+1$ and so $|X| \nless n$ .

$\forall X \subseteq \mathbb{N}\;\exists n \in \mathbb{Z}\;(|X|=n)$ is false, because there exist subsets of $\mathbb{N}$ which are infinite in size, and therefore cannot have cardinality equal to a natural number. For example, $\mathbb{N}$ itself is an infinite subset of $\mathbb{N}$ , as is, for example, the set of all even natural numbers $\{2k \; | \; k \in \mathbb{N}\}$ .

$\forall n \in \mathbb{Z}\;\exists X \subseteq \mathbb{N}\;(|X|=n)$ is false, because the statement breaks for any negative value of $n$ , which is allowed since $\mathbb{Z}$ includes the negative integers. No set can have negative cardinality.

$\forall n \in \mathbb{Z}\; \exists m \in \mathbb{Z}\; (m=n+5)$ is true, because $\mathbb{Z}$ is closed under addition, and $5 \in \mathbb{Z}$ . That is, the set of integers has the property that adding any two integers will always return an integer.

$\exists m \in \mathbb{Z}\; \forall n \in \mathbb{Z}\; (m=n+5)$ is false, because any two different values of $n$ will yield a different $m$ (e.g. $n=0 \implies m=5$ , but $n=1 \implies m=6$ ), so $m=n+5$ for a fixed $m$ cannot hold for even two different values of $n$ , let alone all integer values.

Give a worked solution to the whole question.

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This question is used in the following exam:

Week 5: quantifiers and induction by Daniel Mansfield in Discrete Mathematics.

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Taxonomy: mathcentre

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Taxonomy: Context

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Checkpoint description

Kimia Aksir 4 years, 7 months ago

Daniel Mansfield 7 years, 7 months ago

Error in variable testing condition

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Generated value: `list`

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Match choices with answers

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Daniel's copy of Quantifiers1-

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Kimia Aksir 4 years, 7 months ago

Daniel Mansfield 7 years, 7 months ago

Error in variable testing condition

Error

Warning

Generated value: list

Parts

Match choices with answers

Advanced settings

Generated value: `list`