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Does negating an input negate the output?

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We will compare three changes to implication:

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complement its output, negate X, or negate Y.

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Keeping the original matrix visible lets us explain the differences instead of memorizing names.

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You will learn when to flip cell values and when to permute rows or columns.

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For output negation, turn every zero into one and every one into zero.

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The sole true result is now X equal to one and Y equal to zero.

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Its operator is the upward double arrow, meaning X and not Y.

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For a negated left input, the new X-equals-one row must use the old X-equals-zero row.

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The other row swaps in the opposite direction. On implication this produces OR:

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not X implies Y is equivalent to X or Y.

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For a negated right input, swap the columns.

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The original failure at X one, Y zero is now at X one, Y one.

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This is NAND: X implies not Y is false only when both X and Y are true.

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Let X and Y both be one. Original implication gives one. Negating the output gives zero.

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Negating only X gives one. Negating only Y gives zero.

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Two answers happen to agree here, but the complete matrices distinguish the operations.

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Swap both rows and columns to negate both inputs. For a two-by-two array,

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that has the same effect as a half turn. Implication becomes reverse implication.

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This is the familiar identity: not X implies not Y is equivalent to Y implies X.

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The paper writes some input changes as a transpose followed by a rotation.

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A transpose followed by ninety degrees clockwise swaps columns;

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a transpose followed by two hundred seventy degrees clockwise swaps rows.

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The direct row-and-column rule is usually easier to apply.

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Complement every bit of the exclusive-or matrix. Does
the new rule accept equal inputs or different inputs?

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Name the resulting operator if you remember it.
You can answer entirely from the two-by-two array.

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Complementing exclusive-or produces equivalence, with ones on the diagonal.

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Equal inputs are accepted and different inputs are
rejected. We can now alter inputs or outputs deliberately.

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Next we will combine complete matrices, starting with exclusive-or composition.