Lesson test: Quantifiers

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Quantifiers

1.[2p]
What is the negation of $\forall x > 0 (x^{2} > x)$ , over the reals?

$\exists x > 0 (x^{2} \leq x)$

Correct

$\forall x > 0 (x^{2} \leq x)$

The answer is: $\exists x > 0 (x^{2} \leq x)$

$\exists x \leq 0 (x^{2} \leq x)$

The answer is: $\exists x > 0 (x^{2} \leq x)$

$\exists x \leq 0 (x^{2} > x)$

The answer is: $\exists x > 0 (x^{2} \leq x)$

2.[2p]
The claim " $x^{2} \geq x$ for every $x$ " is true in which of these domains?

Select all that apply

The natural numbers $ℕ$

Correct

The integers $ℤ$

Correct

The rationals $ℚ$

The answer is: The natural numbers $ℕ$ , The integers $ℤ$

The reals $ℝ$

The answer is: The natural numbers $ℕ$ , The integers $ℤ$

3.[1p]
Which symbolises "some prime is even", over the integers?

$\exists n (n is prime \land n is even)$

Correct

$\exists n (n is prime \Rightarrow n is even)$

The answer is: $\exists n (n is prime \land n is even)$

$\forall n (n is prime \land n is even)$

The answer is: $\exists n (n is prime \land n is even)$

$\forall n (n is prime \Rightarrow n is even)$

The answer is: $\exists n (n is prime \land n is even)$

4.[1p]
A claim of the form $\forall x (P (x) \Rightarrow Q (x))$ is refuted by any value of $x$ at which $P (x)$ is false.

True

The answer is: False

False

Correct

5.[2p]
Over the integers, match each sentence to its symbolisation.

Every integer has a larger one
Some integer is larger than every integer
Every integer has a smaller one
Some integer is smaller than every integer

$\forall x \exists y (y < x)$
$\forall x \exists y (y > x)$
$\exists y \forall x (y < x)$
$\exists y \forall x (y > x)$

Show the answer

Every integer has a larger one: $\forall x \exists y (y > x)$ Some integer is larger than every integer: $\exists y \forall x (y > x)$ Every integer has a smaller one: $\forall x \exists y (y < x)$ Some integer is smaller than every integer: $\exists y \forall x (y < x)$

6.[1p]
Over the integers, $\exists y \forall x (y > x)$ is true.

True

The answer is: False

False

Correct

7.[3p]
Which of these statements are true over the integers?

Select all that apply

$\forall x \exists y (y > x)$

Correct

$\forall x \exists y (x + y = 0)$

Correct

$\exists y \forall x (x + y = x)$

Correct

$\exists y \forall x (x y = 0)$

Correct

$\exists y \forall x (y > x)$

The answer is: $\forall x \exists y (y > x)$ , $\forall x \exists y (x + y = 0)$ , $\exists y \forall x (x + y = x)$ , $\exists y \forall x (x y = 0)$

$\forall x \exists y (2 y = x)$

The answer is: $\forall x \exists y (y > x)$ , $\forall x \exists y (x + y = 0)$ , $\exists y \forall x (x + y = x)$ , $\exists y \forall x (x y = 0)$

8.[3p]
A sequence converges to $L$ when $\forall ε > 0 \exists N \in ℕ \forall n \geq N (| a_{n} - L | < ε)$ . Which statement says it does not converge to $L$ ?

$\exists ε > 0 \forall N \in ℕ \exists n \geq N (| a_{n} - L | \geq ε)$

Correct

$\exists ε > 0 \forall N \in ℕ \forall n \geq N (| a_{n} - L | \geq ε)$

The answer is: $\exists ε > 0 \forall N \in ℕ \exists n \geq N (| a_{n} - L | \geq ε)$

$\forall ε > 0 \exists N \in ℕ \forall n \geq N (| a_{n} - L | \geq ε)$

The answer is: $\exists ε > 0 \forall N \in ℕ \exists n \geq N (| a_{n} - L | \geq ε)$

$\exists ε \leq 0 \forall N \in ℕ \exists n \geq N (| a_{n} - L | \geq ε)$

The answer is: $\exists ε > 0 \forall N \in ℕ \exists n \geq N (| a_{n} - L | \geq ε)$

9.[2p]
Put the lines of this proof that $a_{n} = (- 1)^{n}$ does not converge to $- 1$ in order.

We prove the negation of the definition of convergence, with $L = - 1$ .
Let $N$ be any natural number, and take $n = 2 N$ .
Take $ε = 1$ .
Then $n \geq N$ and $n$ is even, so $a_{n} = 1$ and $| a_{n} - (- 1) | = 2 \geq 1$ .
Since $N$ was arbitrary, $(- 1)^{n}$ does not converge to $- 1$ .

Show the answer

a, b, c, d, e

10.[2p]
Which formula says "there is exactly one $x$ with $P (x)$ "?

$\exists x (P (x) \land \forall y (P (y) \Rightarrow y = x))$

Correct

$\exists x P (x) \land \exists y P (y)$

The answer is: $\exists x (P (x) \land \forall y (P (y) \Rightarrow y = x))$

$\forall x \forall y (P (x) \land P (y) \Rightarrow x = y)$

The answer is: $\exists x (P (x) \land \forall y (P (y) \Rightarrow y = x))$

$\exists x \exists y (x \neq y \land P (x) \land P (y))$

The answer is: $\exists x (P (x) \land \forall y (P (y) \Rightarrow y = x))$

11.[3p]
Over the reals, which of these statements are true?

Select all that apply

$\exists! x \forall y (x y = x)$

Correct

$\exists! y \forall x (x y = x)$

Correct

$\forall x \exists! y (x y = x)$

The answer is: $\exists! x \forall y (x y = x)$ , $\exists! y \forall x (x y = x)$ , $\exists! x (x^{3} = - 8)$

$\exists! x (x^{2} = 4)$

The answer is: $\exists! x \forall y (x y = x)$ , $\exists! y \forall x (x y = x)$ , $\exists! x (x^{3} = - 8)$

$\exists! x (x^{3} = - 8)$

Correct

12.[2p]
The definitions of " $f$ is continuous" and " $f$ is uniformly continuous" on $ℝ$ use the same four quantifiers. What is the difference?

In continuity $\forall a$ comes before $\exists δ$ , so $δ$ may depend on the point; in uniform continuity it comes after

Correct

Uniform continuity uses $\leq ε$ where continuity uses $< ε$

The answer is: In continuity $\forall a$ comes before $\exists δ$ , so $δ$ may depend on the point; in uniform continuity it comes after

In uniform continuity $δ$ may not depend on $ε$

The answer is: In continuity $\forall a$ comes before $\exists δ$ , so $δ$ may depend on the point; in uniform continuity it comes after

Uniform continuity requires $δ = ε$

The answer is: In continuity $\forall a$ comes before $\exists δ$ , so $δ$ may depend on the point; in uniform continuity it comes after

13.[2p]
Which statement says that $f : ℝ \to ℝ$ is unbounded?

$\forall M \exists x (| f (x) | > M)$

Correct

$\exists x \forall M (| f (x) | > M)$

The answer is: $\forall M \exists x (| f (x) | > M)$

$\exists M \forall x (| f (x) | > M)$

The answer is: $\forall M \exists x (| f (x) | > M)$

$\forall M \forall x (| f (x) | > M)$

The answer is: $\forall M \exists x (| f (x) | > M)$

14.[1p]
Over an empty domain, $\forall x P (x)$ is true whatever the predicate $P$ is.

True

Correct

False

The answer is: True

1.[2p] What is the negation of ∀x>0(x2>x), over the reals?

2.[2p] The claim "x2≥x for every x" is true in which of these domains?

3.[1p] Which symbolises "some prime is even", over the integers?

4.[1p] A claim of the form ∀x(P(x)⇒Q(x)) is refuted by any value of x at which P(x) is false.

5.[2p] Over the integers, match each sentence to its symbolisation.

6.[1p] Over the integers, ∃y∀x(y>x) is true.

7.[3p] Which of these statements are true over the integers?

8.[3p] A sequence converges to L when ∀ε>0∃N∈ℕ∀n≥N(|an−L|<ε). Which statement says it does not converge to L?

9.[2p] Put the lines of this proof that an=(−1)n does not converge to −1 in order.

10.[2p] Which formula says "there is exactly one x with P(x)"?

11.[3p] Over the reals, which of these statements are true?

12.[2p] The definitions of "f is continuous" and "f is uniformly continuous" on ℝ use the same four quantifiers. What is the difference?

13.[2p] Which statement says that f:ℝ→ℝ is unbounded?

14.[1p] Over an empty domain, ∀xP(x) is true whatever the predicate P is.

1.[2p]
What is the negation of $\forall x > 0 (x^{2} > x)$ , over the reals?

2.[2p]
The claim " $x^{2} \geq x$ for every $x$ " is true in which of these domains?

3.[1p]
Which symbolises "some prime is even", over the integers?

4.[1p]
A claim of the form $\forall x (P (x) \Rightarrow Q (x))$ is refuted by any value of $x$ at which $P (x)$ is false.

5.[2p]
Over the integers, match each sentence to its symbolisation.

6.[1p]
Over the integers, $\exists y \forall x (y > x)$ is true.

7.[3p]
Which of these statements are true over the integers?

8.[3p]
A sequence converges to $L$ when $\forall ε > 0 \exists N \in ℕ \forall n \geq N (| a_{n} - L | < ε)$ . Which statement says it does not converge to $L$ ?

9.[2p]
Put the lines of this proof that $a_{n} = (- 1)^{n}$ does not converge to $- 1$ in order.

10.[2p]
Which formula says "there is exactly one $x$ with $P (x)$ "?

11.[3p]
Over the reals, which of these statements are true?

12.[2p]
The definitions of " $f$ is continuous" and " $f$ is uniformly continuous" on $ℝ$ use the same four quantifiers. What is the difference?

13.[2p]
Which statement says that $f : ℝ \to ℝ$ is unbounded?

14.[1p]
Over an empty domain, $\forall x P (x)$ is true whatever the predicate $P$ is.