 Solution

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/8/a/solution

With all variables ranging over the domain of the partially ordered set, the three first-order logic formulas are

reflexivity: antisymmetry: transitivity: \forall x R (x, x), \forall x \forall y ((R (x, y) \land R (y, x)) \Rightarrow x = y), \forall x \forall y \forall z ((R (x, y) \land R (y, z)) \Rightarrow R (x, z)) .

The middle formula defines an antisymmetric relation; it does not forbid

R (x, y)

for distinct

x, y

, but forbids both directions simultaneously for distinct elements.

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