Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Discrete mathematics, probability and continuous mathematics / 3 / f

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- Solution f

Solution

In a symmetric relation, each of the

n

(a, a)

can be chosen independently. For each of the

(2 n)

unordered pairs of distinct elements, either both ordered pairs

(a, b), (b, a)

occur or neither occurs. Every such choice is free and produces a distinct symmetric relation. Hence

2^{n} 2^{(2 n)} = 2^{n (n + 1) /2} .

For the empty set this correctly gives one binary relation, the empty one.

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