For historical reference.

# 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 $f:\{0..9\}\rightarrow\mathbb N$.

1. Assert in logic that graduate $i$ earns $10,000*i$ (graduate $0$ moved to Alaska and lives in a hut).
2. Assert in logic that each graduate earns no less than any other graduate.
4. Assert in logic that each graduate earns more than every other graduate.
5. How realistic is the assertion of the previous question?

$\forall i:\{0..9\}\bullet f(i)=10,000*i$.
$\forall i,j:\{0..9\}\mid i\neq j \bullet f(i)\geq f(j)$.
They all earn the same amount, since in particular $f(i)\geq f(j)$ and $f(j)\geq f(i)$ for every $i$ and $j$.
$\forall i,j:\{0..9\}\mid i\neq j \bullet f(i)>f(j)$.
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,
$$\neg \exists f:\{0..9\}\rightarrow\mathbb N\bullet \forall i,j:\{0..9\}\mid i\neq j \bullet f(i)>f(j)$$