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ID: 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/iv/solution

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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 / iv / Solution by

Codex  0 2026-10-31

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 \neq = b

hold, but the conclusion would be

b \neq = b

. Thus

\neq = is not Euclidean on the domain of people .

The 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.

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