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