WEBVTT

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When should the door open?

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Suppose X means your pass is valid and Y means access is enabled.

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One means yes, and zero means no. The door opens only when both are true:

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X AND Y. We will turn that rule into a labelled matrix
and learn how to read an answer from the right cell.

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Two yes-or-no inputs give four situations. Both are yes; only X is yes; only Y is yes;

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or neither is yes. A truth table records the answer in every situation, so no case is left out.

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For this door, only the first situation opens it. The other three give zero.

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These four output bits describe the entire rule. A single bit, by itself, describes just one situation.

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Now arrange the same answers in a square. X chooses the row. Y chooses the column.

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Here we use the paper convention: one comes before zero on
both axes. Watch the both-yes answer move to their intersection.

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Suppose your pass is valid, but access is disabled. That means X is one and Y is zero.

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Follow the top row to the right-hand column. The zero there says the door stays closed.

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We put an operator symbol in square brackets to name its correspondence matrix.

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Bracket AND equals the array shown.

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If the operator is not chosen yet, bracket theta stands for its C M.

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Once it is chosen, the symbol and numeric array describe the same ordered object.

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For a different door, either condition might be enough. That is OR.

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The axes stay where they are, but two more answers become one.

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The matrix represents the rule only when we also know what its axes mean.

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How many two-input rules could we make? Each
of the four cells can independently contain zero or one.

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Two choices, four times, gives

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Two choices, four times, gives sixteen complete Boolean functions.

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AND and OR are two members of that family.

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The all-ones matrix is tautological: every input pair produces one.

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Its operator is called Impax. The symbol places the
exclusive-or arrow directly over the equivalence arrow,

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sharing the same center. Agreement and disagreement together cover every possible input pair.

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Try this matrix. Its ones are in the two off-diagonal positions.

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Read the axis labels and decide: does the rule accept inputs that agree, or inputs that differ?

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Take a few seconds to trace both ones.

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It accepts inputs that differ. This is exclusive-or. One and zero gives one;

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zero and one also gives one. Equal inputs give zero.

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Throughout these lessons, the double vertical arrow is our written symbol for exclusive-or.

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Sometimes we need a name for an operator before choosing which one.

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We use theta in brackets for its C M. If we choose exclusive-or,

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bracket theta equals bracket exclusive-or, which equals the off-diagonal numeric matrix.

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Theta is a placeholder, not an additional Boolean rule.

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The project website places zero before one in its introductory example.

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With that ordering, the AND entry is at the bottom right. Nothing about the door has changed.

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Both axes were reordered. Always check the labels before comparing matrices.

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You can now turn a truth table into a matrix and retrieve an answer from it.

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Next we will build whole matrices from four simple pieces.
Keep the row and column labels attached as we do.