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Can two complete rules simplify to one input?

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Take exclusive-or between two complete rules: the implication from X to Y, and the OR of X and Y.

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Both terms use X on the left and Y on the right, so
their matrices can combine at matching coordinates.

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We will find the simpler rule, check all four assignments,
and learn why operand order and negations must be aligned.

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Compute one exclusive-or one at the top left, then zero exclusive-or one at the top right.

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The bottom row similarly becomes zero, one. Both rows are
now identical, so the result cannot depend on the left input X.

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The right input Y is one in the first column and zero in the second.

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The output is zero in the first column and one in the second, so the result is not Y.

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The paper names this matrix bracket not R, for the complemented right projection.

10
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The printed page-twelve example labels this result not
X. Under the declared axes, that is a source discrepancy.

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Take X zero and Y one: the original expression gives zero, while not X gives one.

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Our recomputed matrix and truth table agree on not Y.

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Now change the first expression to X implies not Y.

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Its right operand no longer matches the Y in the OR term.

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Swap the implication matrix columns to absorb the negation.

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Both terms then have the same bra X and ket Y.

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NAND exclusive-or OR has ones on the diagonal and zeros off the diagonal. It is equivalence.

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The paper reaches the same final result, but one intermediate
bracket in its printed chain still shows implication.

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The intermediate array here is explicitly NAND.

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What if the expression is Y implies X, exclusive-or X or Y?

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Rewrite the first term with X as its left selector before combining.

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Which variable survives, and is it negated? Use the transpose rule from lesson ten.

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Reverse implication exclusive-or OR gives zero, zero above one, one.

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That is the complemented left projection, not X.

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Compare it with the earlier not Y result: the difference
follows directly from which implication we started with.