From truth tables to operator CMs

When should the door open? · 0:00

When should the door open? Suppose X means your pass is valid and Y means access is enabled. One means yes, and zero means no. The door opens only when both are true: X AND Y. We will turn that rule into a labelled matrix and learn how to read an answer from the right cell.

List every possible situation · 0:20

Two yes-or-no inputs give four situations. Both are yes; only X is yes; only Y is yes; or neither is yes. A truth table records the answer in every situation, so no case is left out.

Evaluate the rule · 0:34

For this door, only the first situation opens it. The other three give zero. These four output bits describe the entire rule. A single bit, by itself, describes just one situation.

Give each input an axis · 0:46

Now arrange the same answers in a square. X chooses the row. Y chooses the column. Here we use the paper convention: one comes before zero on both axes. Watch the both-yes answer move to their intersection.

Read one cell · 0:58

Suppose your pass is valid, but access is disabled. That means X is one and Y is zero. Follow the top row to the right-hand column. The zero there says the door stays closed.

Give the CM its operator name · 1:10

We put an operator symbol in square brackets to name its correspondence matrix. Bracket AND equals the array shown. If the operator is not chosen yet, bracket theta stands for its C M. Once it is chosen, the symbol and numeric array describe the same ordered object.

Change the rule, keep the axes · 1:29

For a different door, either condition might be enough. That is OR. The axes stay where they are, but two more answers become one. The matrix represents the rule only when we also know what its axes mean.

Why exactly sixteen? · 1:43

How many two-input rules could we make? Each of the four cells can independently contain zero or one. Two choices, four times, gives sixteen complete Boolean functions. AND and OR are two members of that family.

Impax: every input pair is accepted · 2:00

The all-ones matrix is tautological: every input pair produces one. Its operator is called Impax. The symbol places the exclusive-or arrow directly over the equivalence arrow, sharing the same center. Agreement and disagreement together cover every possible input pair.

Your turn: read the pattern · 2:20

Try this matrix. Its ones are in the two off-diagonal positions. Read the axis labels and decide: does the rule accept inputs that agree, or inputs that differ? Take a few seconds to trace both ones.

Prediction pause: 8 seconds.

Answer: the inputs differ · 2:43

It accepts inputs that differ. This is exclusive-or. One and zero gives one; zero and one also gives one. Equal inputs give zero. Throughout these lessons, the double vertical arrow is our written symbol for exclusive-or.

Theta stands for a chosen operator · 3:00

Sometimes we need a name for an operator before choosing which one. We use theta in brackets for its C M. If we choose exclusive-or, bracket theta equals bracket exclusive-or, which equals the off-diagonal numeric matrix. Theta is a placeholder, not an additional Boolean rule.

Same rule, different ordering · 3:22

The project website places zero before one in its introductory example. With that ordering, the AND entry is at the bottom right. Nothing about the door has changed. Both axes were reordered. Always check the labels before comparing matrices.

What you can now do · 3:37

You can now turn a truth table into a matrix and retrieve an answer from it. Next we will build whole matrices from four simple pieces. Keep the row and column labels attached as we do.