Foundations — Sum of products (SOP) form

Before you can read the parent note on SOP form, you must own every symbol it throws at you. This page builds each one from absolute zero, in the order they depend on each other. Nothing is used before it is drawn.

0. The atom: a bit (0 and 1)

The picture: a light switch. It is either OFF ($0$) or ON ($1$). That is the entire universe we work in.

A variable like $A$, $B$, $C$ is just a named box that holds one bit. When we say "$A=1$" we mean "the box called $A$ currently holds ON".

1. NOT — the flip ( $\bar{A}$ )

Why the topic needs it: SOP terms must be able to say "this input was OFF". The only way to write "$A$ was $0$" as something that equals 1 is to flip it: $\bar{A}$ equals $1$ exactly when $A=0$. That flip is the "zero gets a hat" rule in the parent note.

The picture: a switch wired to an inverter — press ON, the bulb goes OFF.

2. AND — everybody must agree ( $\cdot$ , written by touching )

The picture: two switches wired in series (one after the other). Current reaches the bulb only if switch 1 AND switch 2 are both closed. Break either one and the bulb dies.

Why the topic needs it: each "row-detector" in SOP is an AND of literals. AND is the tool that answers the question "were ALL of these conditions true at once?" — no other single operation asks that.

3. OR — anybody will do ( $+$ )

The picture: two switches wired in parallel (side by side). Current has two paths to the bulb, so closing either switch lights it. Both open = dark.

Why the topic needs it: SOP glues the row-detectors together with OR. OR answers the question "did ANY winning combination happen?" — which is exactly what "the output is 1 on these rows" means.

4. Literal — a variable in either dress

Think of it as a switch you can wire "normally" (fires when pressed) or "inverted" (fires when not pressed). A product term is built entirely out of literals.

5. Truth table — the full behaviour chart

The picture: an odometer counting in binary. Two variables → $2^2=4$ rows: $00,01,10,11$. Three variables → $2^3=8$ rows.

Why the topic needs it: SOP is read directly off this table — you pick the rows where the output is $1$. See Truth Tables for the full construction.

6. Putting the symbols in order — the minterm

Now that NOT, AND, OR, literal and truth table exist, the parent's central object costs nothing:

The $\sum m$ shorthand

The large $\sum$ (sigma) is not arithmetic summation here — it means "OR all of these together", matching part 3.

How every foundation feeds SOP

Read it top-down: bits make variables; NOT and AND turn variables into minterms; the truth table tells you which minterms to build; OR joins them into SOP.

Equipment checklist

Test yourself — reveal only after answering aloud.

Connections

• Parent: Sum of Products form — this page equips you for it.
• Logic Gates (AND, OR, NOT) — the physical devices behind these three symbols.
• Truth Tables — where SOP is read from.
• Minterms and Maxterms — the row-detector, formalised.
• Boolean Algebra Laws — rules for simplifying once you have SOP.
• Product of Sums (POS) form — the mirror-image method.
• Karnaugh Maps — visual simplification.