Chapter 0 – Quiz 1 – E600 Mathematics

1. What does the following statement mean verbally: “ $\forall n\in\mathbb N: (\forall x\in\mathbb R: (x > 0 \lor n > x))$ “? (Hint: the statement is meaningful and true.)

For all integers $n$ and all elements $x$ of the rational numbers, $x$ is greater than zero and $n$ is greater than $x$ .

There exist a natural number $n$ and a rational number $x$ so that $x$ is strictly greater than zero and $n$ is strictly greater than $x$ .

For all elements $n$ of the natural numbers and all elements $x$ of the real numbers, $x$ is strictly greater than zero and $n$ is strictly greater than $x$ .

For all elements $n$ of the natural numbers and all elements $x$ of the real numbers, $x$ is strictly greater than zero or $n$ is strictly greater than $x$ .

2. Which of the following statements are true? (Multiple choice)

$\forall x\in\mathbb R: (\forall y\in\mathbb R: x + y< 0)$

$\forall x\in\mathbb R: x < 0$

$\exists x\in\mathbb R: x < 0$

$\nexists n\in\mathbb N: n - 2 < 0$

$\forall n\in\mathbb N: n\geq 0$

$\exists x\in\mathbb R: (\exists y\in\mathbb R: x + y< 0)$

3. Find the statement that expresses the exact opposite of $\exists n\in\mathbb N: n+1 < 0$ , i.e. find the negation $\neg(\exists n\in\mathbb N: n+1 < 0)$ .