# Tutorial II

## The questions

Ten graduates called $0$ to $9$ assemble in a reunion to compare their salaries. Represent the salary by a number $n:\mathbb N$ and represent the graduates’ salaries as a function $f:\{0..9\}\rightarrow\mathbb N$.

- Assert in logic that graduate $i$ earns $10,000*i$ (graduate $0$ moved to Alaska and lives in a hut).
- Assert in logic that each graduate earns no less than any other graduate.
- What does that tell you about what the graduates earn?
- Assert in logic that each graduate earns more than every other graduate.
- How realistic is the assertion of the previous question?

## The answers (contains spoilers)

- Click here to reveal

$\forall i:\{0..9\}\bullet f(i)=10,000*i$. - Click here to reveal

$\forall i,j:\{0..9\}\mid i\neq j \bullet f(i)\geq f(j)$. - Click here to reveal

They all earn the same amount, since in particular $f(i)\geq f(j)$ and $f(j)\geq f(i)$ for every $i$ and $j$. - Click here to reveal

$\forall i,j:\{0..9\}\mid i\neq j \bullet f(i)>f(j)$. - Click here to reveal

Impossible, since in particular $f(0)>f(1)$ and $f(1)>f(0)$. The assertion is still a perfectly valid predicate; there is just no $f$ that satisfies it. In other words,

$$

\neg \exists f:\{0..9\}\rightarrow\mathbb N\bullet \forall i,j:\{0..9\}\mid i\neq j \bullet f(i)>f(j)

$$

is true. Simples!