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Sums of truth-table values mod 2 range over all truth tables

Let $A=\lbrace0,1\rbrace$. There are 16 distinct functions $f_i:A^2\to A$. Choose a permutation $P=\left(a_1,\ldots,a_4\right)$ of the elements of $A^2$, and for each $i$ consider the ordered ...