Past exam of the year 1 of the computer science course of the University of Oxford / 2003 / Calculus, probability, linear algebra, discrete mathematics, logic and proof, and models of computation / 6 / a

The binary relation $x\le y$ on the natural numbers is not a Euclidean relation. Choose $x=1$, $y=3$, $z=2$. Then $1\le 3$ and $1\le 2$, but $3\nleqq 2$. Thus

The divisibility binary relation on positive natural numbers is not a Euclidean relation. With $x=1$, $y=2$, $z=3$, we have $1\mid 2$ and $1\mid 3$ but $2\nmid 3$. Therefore

Equality of remainders after division by eight is a Euclidean relation. If $x$, $y$ have the same remainder and $x$, $z$ have the same remainder, then $y$, $z$ have the same remainder. In the notation of modular arithmetic,Hence

Inequality on a domain containing at least two people is not a Euclidean relation. Choose distinct people $a,b$ and put $x=a$, $y=z=b$. Both premises $a\ne b$ hold, but the conclusion would be $b\ne b$. ThusThe possibility $y=z$ matters: the defining universal quantifiers do not require distinct arguments. On a domain with at most one element the inequality binary relation is empty and the condition holds by vacuous truth.

The “same age” binary relation is a Euclidean relation, with age understood as one consistently measured value at a fixed time. If $x$ has the same age as both $y$ and $z$, the ages of $y$ and $z$ are equal. HenceThis is another instance of equality of the values of a function inducing an equivalence relation.