Boolean Foundations on Digital Logic Notebook

Truth Tables

Mon, 01 Jan 0001 00:00:00 +0000

Truth Tables#

A truth table is the complete, unambiguous specification of a combinational function: every possible input combination on the left, the output it must produce on the right. It says what a function does without saying anything about how it is built, which makes it the natural starting point for every design and the common ground between a specification and a circuit.

Complete, but It Scales Badly#

For n inputs there are 2ⁿ combinations, so a truth table has 2ⁿ rows. That is fine for the two- and three-input gates whose behavior is a small truth table, but it grows explosively: four inputs give 16 rows, eight give 256, and twenty give over a million. Completeness is the strength and the weakness — nothing is left unspecified, but the table is unusable as a design tool much past a handful of variables. That is exactly why the rest of this subsection exists: Boolean algebra and canonical forms compress the table into an expression, and Karnaugh maps reorganize it to be simplified by eye.

Boolean Algebra

Mon, 01 Jan 0001 00:00:00 +0000

Boolean Algebra#

Boolean algebra is the algebra of two-valued logic: variables that are only ever 0 or 1, combined with three operations — AND (·), OR (+), and NOT (′). It is the lever between a truth table and a circuit, because it lets a logic expression be manipulated and simplified on paper before a single gate is committed. Fewer terms mean fewer gates, fewer levels of logic, less delay, and less power.

De Morgan's Theorems

Mon, 01 Jan 0001 00:00:00 +0000

De Morgan’s Theorems#

De Morgan’s theorems are the two most-used identities in all of digital logic. They describe what happens when the output of an AND or an OR is inverted:

$$ (A \cdot B)’ = A’ + B’ $$

$$ (A + B)’ = A’ \cdot B’ $$

In words: inverting the output of a gate is the same as swapping the gate type (AND ↔ OR) and inverting every input. The theorems extend to any number of inputs — the complement of a whole product is the sum of the complemented terms, and vice versa.

Canonical Forms (SOP & POS)

Mon, 01 Jan 0001 00:00:00 +0000

Canonical Forms (SOP & POS)#

A canonical form is a systematic, mechanical way to write down a function straight from its truth table — no cleverness required. There are two, and they are duals of each other: sum of products, built from the rows where the output is 1, and product of sums, built from the rows where it is 0.

Sum of Products (from the 1s)#

For each row where the output is 1, write a minterm: the AND of every input variable, taken true where the variable is 1 and complemented where it is 0. OR all those minterms together and the result is the sum-of-products (SOP) form — it is 1 for exactly the input combinations that should give 1, and nothing else. For a function that is 1 on rows 3 and 5 of three variables, F = A′BC + AB′C, often written compactly as Σm(3, 5).

Karnaugh Maps

Mon, 01 Jan 0001 00:00:00 +0000

Karnaugh Maps#

A Karnaugh map is a truth table rearranged so that simplification becomes something the eye can do. The rows and columns are ordered in Gray code — each cell differs from its neighbors by exactly one variable — so that physically adjacent 1s can always be combined into a simpler term. It turns the algebra of minimization into pattern-spotting.

Reading the Grid#

Here is a three-variable function, F(A, B, C), with the columns in Gray order (00, 01, 11, 10):

Hazards & Glitches

Mon, 01 Jan 0001 00:00:00 +0000

Hazards & Glitches#

A truth table describes what a function settles to, not the momentary values it passes through on the way. Real gates take time to switch, and different signal paths through a circuit take different amounts of time — so when an input changes, the output can briefly show a wrong value before settling to the right one. That transient is a glitch, and the circuit property that permits it is a hazard. The logic is correct; the timing is not.