Canonical forms — SOP & POS

How to write a Boolean function straight from its truth table, two standard ways. This is the starting expression every simplification begins from.

Built from first principles.

Before you start

What you need first

Truth tables — every row, in binary order.
Theorems & simplification — why a smaller expression is better.

What you'll be able to do

Write the SOP form from the rows where F = 1.
Write the POS form from the rows where F = 0.
Use minterm/maxterm and the Σm / ΠM notation.

First we need a starting expression

A truth table says what a function does, but you can't wire a truth table — you wire an expression. Two standard ("canonical") forms come straight out of the table, and either can then be simplified (next topic) into fewer gates.

A simpler expression = fewer gates = smaller, cheaper, cooler, faster. But you have to start somewhere — that's what SOP and POS give you.

Sum-of-Products (SOP): read the 1-rows

A minterm is an AND term using every variable exactly once (itself or complemented). There are $2^n$ of them. Rule: for each row where F = 1, write the minterm true on that row, then OR them:

variable is 1 in that row → write it as itself (A),
variable is 0 in that row → write its complement (A′).

📐 Worked example

Build the SOP for this function

Row	A	B	C	F
0	0	0	0	0
1	0	0	1	1
2	0	1	0	0
3	0	1	1	1
4	1	0	0	0
5	1	0	1	1
6	1	1	0	0
7	1	1	1	1

One minterm per F = 1 row:

$$ \begin{aligned} F &= A'B'C + A'BC \\ &+ AB'C + ABC \\ &= \Sigma m(1,3,5,7) \end{aligned} $$

Every minterm has C as itself — a hint the simplified form is just C.

Product-of-Sums (POS): read the 0-rows

A maxterm is an OR term using every variable once — the opposite of a minterm (false on exactly one row). Rule: for each row where F = 0, write the maxterm false on that row, then AND them:

variable is 1 in that row → write its complement (A′),
variable is 0 in that row → write it as itself (A).

📐 Worked example

Same function, POS from the F = 0 rows (0, 2, 4, 6)

$$ \begin{aligned} F &= (A+B+C)\,(A+B'+C) \\ &\quad (A'+B+C)\,(A'+B'+C) \\ &= \Pi M(0,2,4,6) \end{aligned} $$

Both forms describe the same function (both simplify to F = C). Choose SOP when 1s are rare, POS when 0s are rare — fewer terms to write and simplify.

✏️ Try it yourself

(a) How many minterms for a 4-variable function? (b) For row A=1,B=0,C=1 with F=1, write the minterm. (c) Same row with F=0, write the maxterm. (d) A function has only one F=0 row — SOP or POS?

(a) $2^4 = 16$. (b) A=1→A, B=0→B′, C=1→C: A·B′·C. (c) flip each: A′ + B + C′. (d) POS — just one maxterm (POS is shorter when 0s are rare).

Recap — the whole topic on one screen

Form	Read from	Term rule	Notation
SOP	F = 1 rows	AND (1→self, 0→comp), then OR	Σm(…)
POS	F = 0 rows	OR (1→comp, 0→self), then AND	ΠM(…)

Pick the one with fewer terms; both are the same function, ready to simplify.

Next topic

Karnaugh maps

SOP straight from the table can be huge. The K-map is a visual way to find the smallest equivalent SOP — no algebra, just fill the grid and circle the groups.

→ Karnaugh maps