For historical reference.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.
  3. What does that tell you about what the graduates earn?
  4. Assert in logic that each graduate earns more than every other graduate.
  5. How realistic is the assertion of the previous question?

The answers (contains spoilers)

  1. Click here to reveal
    $\forall i:\{0..9\}\bullet f(i)=10,000*i$.
  2. Click here to reveal
    $\forall i,j:\{0..9\}\mid i\neq j \bullet f(i)\geq f(j)$.
  3. 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 $j$.
  4. Click here to reveal
    $\forall i,j:\{0..9\}\mid i\neq j \bullet f(i)>f(j)$.
  5. 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!
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