WEBVTT

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Does X implying Y mean that Y implies X?

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Not in general. We will reverse the operands and learn
exactly which matrix entries move under a transpose.

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Keep the true-first convention: the first row and first column mean one.

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The axis labels will explain which question each matrix answers.

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Implication fails only when its left input is one and its right input is zero.

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That is the upper-right cell. For X equal to one and Y equal to zero,

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X implies Y is false, while Y implies X is true.

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A transpose exchanges the row and column indices
of every entry. Diagonal entries stay where they are.

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The upper-right zero moves to the lower-left position.

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We have now made the matrix of reverse implication
under the original X-row and Y-column labels.

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There are two ways to ask the reversed question. We can put Y
on the left of the original implication matrix and X on its right.

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Or keep X on the left and Y on the right, while transposing the matrix.

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Both ask whether Y implies X.

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Use X equal to one and Y equal to zero again.

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With X on the left of the reverse-implication matrix, the bra selects its first row, one, one.

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Ket zero selects the second entry. The result is one, agreeing with zero implies one.

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Exclusive-or is different: its matrix equals its transpose. The two off-diagonal entries are both one.

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This tells us that swapping the operands preserves the exclusive-or rule for every input pair.

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Symmetry is visible in the numeric matrix.

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Transpose the displayed implication matrix once more. Track the zero from its current position,

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then name the bracketed operator you obtain and explain
why every entry returns to its original coordinate.

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The zero moves from lower left back to upper right,
so two transposes recover the implication matrix.

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Every entry returns to its starting coordinates.

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This also explains why reversing the input order twice recovers the original question.

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A matrix transformation has to travel with a statement about the operands and axes.

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Moving entries under fixed axes can change the represented rule.

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Moving entries and relabelling the inputs together can preserve a measurement.

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Next we distinguish input negation from output negation.