If I define a ternary logical connective $\clubsuit(a,b,c)$ by the following truth table: \begin{array}{ccc|c}a&b&c&\clubsuit(a,b,c)\\\hline T&T&T&F\\T&T&F&F\\T&F&T&F\\T&F&F&T\\F&T&T&F\\F&T&F&T\\F&F&T&T\\F&F&F&T\end{array} (whatever it means, $\clubsuit(a,b,c)$ is true when at most one of $\left\{a,b,c\right\}$ is true), how do I determine whether or not the set $\left\{\clubsuit\right\}$ is functionally complete?
2026-04-13 11:46:08.1776080768
Functional completeness for a ternary operator
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Sorry I said symmetric, it is a translation error. In English this property is known as self-dual: function is self dual if $f(x) = \neg{f(\neg{x})}$, which is the case for your function. A complete set must contain a non-self-dual function.
The self-duality is inherited under composition: any composition of self-dual functions is self-dual as well. Therefore it is not possible to express a not self-dual function.