 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/6/b/iv/solution

On the nonempty set

X = {a}

take the empty binary relation

R = \emptyset

. There are no triples for which

x R y \land x R z

holds. The defining implication is therefore true for every triple by vacuous truth, making

R

But

a R a

is false, so

R

(X, R) = ({a}, \emptyset) is a counterexample .

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