Lesson test: Relations and multiple quantifiers

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Relations and multiple quantifiers

1.[3p]
In the domain of positive integers with $G x y$ meaning $x > y$ , which is true?

$\forall x \exists y G y x$

Correct

$\exists y \forall x G y x$

The answer is: $\forall x \exists y\, Gyx$

$\forall x \forall y G y x$

The answer is: $\forall x \exists y\, Gyx$

$\exists y \forall x G x y$

The answer is: $\forall x \exists y\, Gyx$

2.[3p]
Why is $\exists y \forall x R x y$ stronger than $\forall x \exists y R x y$ ?

The $y$ must be fixed before $x$ is considered, so one $y$ has to serve every $x$

Correct

It uses fewer variables

The answer is: The $y$ must be fixed before $x$ is considered, so one $y$ has to serve every $x$

The existential quantifier is always stronger

The answer is: The $y$ must be fixed before $x$ is considered, so one $y$ has to serve every $x$

It has no free variables

The answer is: The $y$ must be fixed before $x$ is considered, so one $y$ has to serve every $x$

3.[2p]
$\neg \forall x F x$ is equivalent to

$\exists x \neg F x$

Correct

$\forall x \neg F x$

The answer is: $\exists x\, \neg Fx$

$\neg \exists x F x$

The answer is: $\exists x\, \neg Fx$

$\exists x F x$

The answer is: $\exists x\, \neg Fx$

4.[3p]
Negating $\forall x (S x \to \exists y (B y \land R x y))$ gives

$\exists x (S x \land \forall y (B y \to \neg R x y))$

Correct

$\exists x (S x \to \forall y (B y \land \neg R x y))$

The answer is: $\exists x (Sx \wedge \forall y (By \rightarrow \neg Rxy))$

$\forall x (S x \land \exists y (B y \to \neg R x y))$

The answer is: $\exists x (Sx \wedge \forall y (By \rightarrow \neg Rxy))$

$\exists x (\neg S x \land \forall y (B y \to R x y))$

The answer is: $\exists x (Sx \wedge \forall y (By \rightarrow \neg Rxy))$

5.[3p]
Match each property of a relation $R$ to its formula.

Reflexive
Symmetric
Transitive

$\forall x \forall y \forall z ((R x y \land R y z) \to R x z)$
$\forall x R x x$
$\forall x \forall y (R x y \to R y x)$

Show the answer

Reflexive: $\forall x R x x$ Symmetric: $\forall x \forall y (R x y \to R y x)$ Transitive: $\forall x \forall y \forall z ((R x y \land R y z) \to R x z)$

6.[2p]
$\forall x \forall y R x y$ and $\forall y \forall x R x y$ say the same thing.

True

Correct

False

The answer is: True

7.[3p]
Continuity and uniform continuity differ in that

uniform continuity moves $\exists δ$ in front of $\forall x$ , so one $δ$ must work everywhere

Correct

uniform continuity uses a different inequality

The answer is: uniform continuity moves $\exists \delta$ in front of $\forall x$, so one $\delta$ must work everywhere

continuity quantifies over $ε$ and uniform continuity does not

The answer is: uniform continuity moves $\exists \delta$ in front of $\forall x$, so one $\delta$ must work everywhere

uniform continuity applies only to bounded functions

The answer is: uniform continuity moves $\exists \delta$ in front of $\forall x$, so one $\delta$ must work everywhere

8.[2p]
For $f (x) = x^{2}$ , what is ${1000.001}^{2} - 1000^{2}$ , to three decimal places?

CorrectNot quite: 2.000

9.[3p]
"Every lock has a key that opens it" and "there is a key that opens every lock" are related how?

The second entails the first, and not the other way round

Correct

The first entails the second, and not the other way round

The answer is: The second entails the first, and not the other way round

They are equivalent

The answer is: The second entails the first, and not the other way round

Neither entails the other

The answer is: The second entails the first, and not the other way round

1.[3p] In the domain of positive integers with Gxy meaning x>y, which is true?

2.[3p] Why is ∃y∀xRxy stronger than ∀x∃yRxy?

3.[2p] ¬∀xFx is equivalent to

4.[3p] Negating ∀x(Sx→∃y(By∧Rxy)) gives

5.[3p] Match each property of a relation R to its formula.

6.[2p] ∀x∀yRxy and ∀y∀xRxy say the same thing.

7.[3p] Continuity and uniform continuity differ in that

8.[2p] For f(x)=x2, what is 1000.0012-10002, to three decimal places?

9.[3p] "Every lock has a key that opens it" and "there is a key that opens every lock" are related how?

1.[3p]
In the domain of positive integers with $G x y$ meaning $x > y$ , which is true?

2.[3p]
Why is $\exists y \forall x R x y$ stronger than $\forall x \exists y R x y$ ?

3.[2p]
$\neg \forall x F x$ is equivalent to

4.[3p]
Negating $\forall x (S x \to \exists y (B y \land R x y))$ gives

5.[3p]
Match each property of a relation $R$ to its formula.

6.[2p]
$\forall x \forall y R x y$ and $\forall y \forall x R x y$ say the same thing.

7.[3p]
Continuity and uniform continuity differ in that

8.[2p]
For $f (x) = x^{2}$ , what is ${1000.001}^{2} - 1000^{2}$ , to three decimal places?

9.[3p]
"Every lock has a key that opens it" and "there is a key that opens every lock" are related how?