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: $\exists x > 0\, (x^2 \le x)$

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

The answer is: $\exists x > 0\, (x^2 \le x)$

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

The answer is: $\exists x > 0\, (x^2 \le x)$

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: $\exists n\, (n \ \text{is prime} \wedge n \ \text{is even})$

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

The answer is: $\exists n\, (n \ \text{is prime} \wedge n \ \text{is even})$

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

The answer is: $\exists n\, (n \ \text{is prime} \wedge n \ \text{is even})$

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: $\forall x\, \exists y\, (y > x)$, $\forall x\, \exists y\, (x + y = 0)$, $\exists y\, \forall x\, (x + y = x)$, $\exists y\, \forall x\, (xy = 0)$

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

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

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: $\exists \varepsilon > 0\ \forall N \in ℕ\ \exists n \ge N\ (|a_n - L| \ge \varepsilon)$

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

The answer is: $\exists \varepsilon > 0\ \forall N \in ℕ\ \exists n \ge N\ (|a_n - L| \ge \varepsilon)$

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

The answer is: $\exists \varepsilon > 0\ \forall N \in ℕ\ \exists n \ge N\ (|a_n - L| \ge \varepsilon)$

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

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.

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.