WEBVTT

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How can larger rules combine without mixing inputs?

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Our rule takes W exclusive-or X, implying the conjunction of NOT Y and Z.

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Rows use W Y and columns use X Z, both true-first.

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We will check a cell against the original rule,

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then learn to permute and combine larger matrices while keeping their assignments aligned.

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Let W be one, X zero, Y zero, and Z one.

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The antecedent is one exclusive-or zero, which is
one. The consequent is not zero and one, also one.

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Implication gives one. Row W Y is one zero and column X Z is zero one.

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Now display rows as Y W instead of W Y. Under true-first binary order,

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the middle two row addresses exchange places. New row zero-one reads old row one-zero.

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Columns remain X Z. This changes the presentation,
provided we change the labels with the entries.

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For the second rule, first form Y AND NOT Z. Compare
that conjunction for equivalence with NOT W,

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then negate the entire equivalence. Negated equivalence is exclusive-or,

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so G is the conjunction Y AND NOT Z, exclusive-or NOT W.

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X is absent, but we still use rows W Y and columns X Z.

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To compute F implies G, apply implication to corresponding cells of their aligned matrices.

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At our chosen assignment, F is one and G is zero, so the combined result is zero.

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This operation is entrywise implication, not Boolean matrix multiplication.

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The completed matrix has one entry for each assignment of W, X, Y, and Z.

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We independently evaluate both original formulas,
then their implication, for all sixteen assignments.

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This also catches a row-order mismatch that one convenient example could miss.

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For the supplied assignment W one, X zero, Y one, Z one,

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calculate F and G, then apply the outer implication.
The component values are shown as a guided check:

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explain how the outer implication uses them and locate the corresponding row and column.

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The first rule is false because its antecedent is true and its consequent is false.

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The second rule is also false at this assignment.

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False implies false is true, so the combined cell is one. This agrees with the full matrix.