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Reinventing the wheel
It seems like I've been trying to reinvent the wheel with my "tricky exercises." Basically, all I needed to do was to count 2-valued functions.
That is, having two "no-brainer" 2-valued functions tells us that there are two parts in the set.
upd. The tricky aspect is restricting the set of functions to two by eliminating {0,0} and {1,1} choices.
upd1. Any two-dimensional choice space is prone to tricky either-or questions. E.g. System 1 vs System 2, short- vs long-term, шашечки или ехать, quality vs price, etc. Under the circumstances, we should start by expanding the choices by expanding from product AxB to disjoint union A|_|B.
The preceding principle will have a great many uses. It enables us, for example, to count parts by counting 2-valued functions instead.
-- Lawvere, Rosebrugh. 2006.
That is, having two "no-brainer" 2-valued functions tells us that there are two parts in the set.
upd. The tricky aspect is restricting the set of functions to two by eliminating {0,0} and {1,1} choices.
upd1. Any two-dimensional choice space is prone to tricky either-or questions. E.g. System 1 vs System 2, short- vs long-term, шашечки или ехать, quality vs price, etc. Under the circumstances, we should start by expanding the choices by expanding from product AxB to disjoint union A|_|B.