Lesson 18 of 23 · 2:44
Transform and combine larger CMs
Align axes, apply the outer rule entrywise, and verify all assignments.
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In this lesson
- 0:00 · How can larger rules combine without mixing inputs?
- 0:24 · Check one cell against the formula
- 0:43 · Permute the row-bit order
- 1:03 · Introduce a second rule in the same layout
- 1:26 · Combine matching assignments
- 1:46 · Verify all sixteen outputs
- 2:03 · Guided consistency check: outer implication
- 2:30 · Answer: false implies false is true
Complete transcript
This transcript includes the answers to prediction prompts.
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How can larger rules combine without mixing inputs? · 0:00
How can larger rules combine without mixing inputs? Our rule takes W exclusive-or X, implying the conjunction of NOT Y and Z. Rows use W Y and columns use X Z, both true-first. We will check a cell against the original rule, then learn to permute and combine larger matrices while keeping their assignments aligned.
Check one cell against the formula · 0:24
Let W be one, X zero, Y zero, and Z one. The antecedent is one exclusive-or zero, which is one. The consequent is not zero and one, also one. Implication gives one. Row W Y is one zero and column X Z is zero one.
Permute the row-bit order · 0:43
Now display rows as Y W instead of W Y. Under true-first binary order, the middle two row addresses exchange places. New row zero-one reads old row one-zero. Columns remain X Z. This changes the presentation, provided we change the labels with the entries.
Introduce a second rule in the same layout · 1:03
For the second rule, first form Y AND NOT Z. Compare that conjunction for equivalence with NOT W, then negate the entire equivalence. Negated equivalence is exclusive-or, so G is the conjunction Y AND NOT Z, exclusive-or NOT W. X is absent, but we still use rows W Y and columns X Z.
Combine matching assignments · 1:26
To compute F implies G, apply implication to corresponding cells of their aligned matrices. At our chosen assignment, F is one and G is zero, so the combined result is zero. This operation is entrywise implication, not Boolean matrix multiplication.
Verify all sixteen outputs · 1:46
The completed matrix has one entry for each assignment of W, X, Y, and Z. We independently evaluate both original formulas, then their implication, for all sixteen assignments. This also catches a row-order mismatch that one convenient example could miss.
Guided consistency check: outer implication · 2:03
For the supplied assignment W one, X zero, Y one, Z one, calculate F and G, then apply the outer implication. The component values are shown as a guided check: explain how the outer implication uses them and locate the corresponding row and column.
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Answer: false implies false is true · 2:30
The first rule is false because its antecedent is true and its consequent is false. The second rule is also false at this assignment. False implies false is true, so the combined cell is one. This agrees with the full matrix.