WEBVTT

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How do two small matrices become a larger one?

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Suppose one complete logical matrix uses W and X, and another uses Y and Z.

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A tensor product keeps every combination of their
entries. We will build the entire four-by-four grid

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one block at a time, then use its row and column components to explain a selected cell.

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M sub W X is the whole matrix on the left of the tensor symbol.

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M sub Y Z is the whole matrix on the right. The equals
signs connect each name to its four expressions.

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W X inside the first matrix is only one entry.

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For an ordinary block expansion, take each entry of the first
matrix and combine it with every entry of the second.

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There are two block rows and two block columns.

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Each block has two rows and two columns, giving four of each overall.

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Begin with the first block. Every expression there includes W and X.

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The second matrix contributes Y and Z, Y and not Z, not Y and Z, or neither.

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These four combinations occupy a square, with two entries in each row.

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The next block uses W and not X. The two lower blocks use not W.

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In each case the second matrix contributes all four of its expressions. Now the complete grid

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contains sixteen expressions, covering all combinations of the four variables.

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Why do the rows use W and Y?

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Each row combines a row choice from the first matrix with a row choice from the second.

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Each column similarly combines X and Z. The row components
and column components are written separately above the grid.

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Follow row W and not Y, then column not X and Z.

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The intersection contains W, not X, not Y, and Z, joined by AND.

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The axis components tell you every factor in that expression without memorising the grid.

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Find the last row and the last column. Which of the four variables are negated at that intersection?

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Read the row components first, then the column
components. You can pause here if you want more time.

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All four variables are negated. The row contributes not W and not Y.

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The column contributes not X and not Z. Joined together, they give the final expression shown.

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This is one entry of the full tensor, not the tensor itself.

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This base tensor organises conjunction expressions.
To construct a more interesting compound rule,

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we must supply the appropriate logical matrices and combine them according to that rule.

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The next lesson does that for exclusive-or followed by implication.