Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 128 / 1 / a

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- Solution a

Solution

First form the singleton set

{b} = F_{1} (b, b)

and then use

F_{2}

to take a set union:

U (a, b) = F_{2} (F_{1} (a, F_{1} (b, b)), a) = ⋃ {a, {b}} = a \cup {b} .

The unused second argument of

F_{2}

may be any term. Since

x \cap c = x ∖ (x ∖ c)

, a term using only the prescribed operation symbols is

G (a, b, c) = F_{3} (U (a, b), F_{3} (U (a, b), c)) .

After substituting the displayed term for both occurrences of

U

, this is literally a term in

F_{1}, F_{2}, F_{3}

, and its value is

(a \cup {b}) \cap c

.

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