Lesson 14 of 23 · 2:36
Combine rules with a shared ordered basis
Normalize order and negations, then compose and verify the result.
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In this lesson
- 0:00 · Can two complete rules simplify to one input?
- 0:25 · XOR the aligned arrays
- 0:39 · Read the remaining right-input rule
- 0:57 · Check the manuscript’s printed projection
- 1:17 · Normalize a negated right operand
- 1:34 · Then combine the normalized rules
- 1:52 · Your turn: reverse the first implication
- 2:18 · Answer: the left input is complemented
Complete transcript
This transcript includes the answers to prediction prompts.
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Can two complete rules simplify to one input? · 0:00
Can two complete rules simplify to one input? Take exclusive-or between two complete rules: the implication from X to Y, and the OR of X and Y. Both terms use X on the left and Y on the right, so their matrices can combine at matching coordinates. We will find the simpler rule, check all four assignments, and learn why operand order and negations must be aligned.
XOR the aligned arrays · 0:25
Compute one exclusive-or one at the top left, then zero exclusive-or one at the top right. The bottom row similarly becomes zero, one. Both rows are now identical, so the result cannot depend on the left input X.
Read the remaining right-input rule · 0:39
The right input Y is one in the first column and zero in the second. The output is zero in the first column and one in the second, so the result is not Y. The paper names this matrix bracket not R, for the complemented right projection.
Check the manuscript’s printed projection · 0:57
The printed page-twelve example labels this result not X. Under the declared axes, that is a source discrepancy. Take X zero and Y one: the original expression gives zero, while not X gives one. Our recomputed matrix and truth table agree on not Y.
Normalize a negated right operand · 1:17
Now change the first expression to X implies not Y. Its right operand no longer matches the Y in the OR term. Swap the implication matrix columns to absorb the negation. Both terms then have the same bra X and ket Y.
Then combine the normalized rules · 1:34
NAND exclusive-or OR has ones on the diagonal and zeros off the diagonal. It is equivalence. The paper reaches the same final result, but one intermediate bracket in its printed chain still shows implication. The intermediate array here is explicitly NAND.
Your turn: reverse the first implication · 1:52
What if the expression is Y implies X, exclusive-or X or Y? Rewrite the first term with X as its left selector before combining. Which variable survives, and is it negated? Use the transpose rule from lesson ten.
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Answer: the left input is complemented · 2:18
Reverse implication exclusive-or OR gives zero, zero above one, one. That is the complemented left projection, not X. Compare it with the earlier not Y result: the difference follows directly from which implication we started with.