Foundations — Product of sums (POS) form

This page builds every symbol the POS note leans on — from the digit $0$ all the way to the giant $\prod$ — with a picture for each. If a word in the parent note ever made you pause, its meaning is built here, in order, from nothing.

1. The two values: $0$ and $1$

We also read them as words: $1=$ true / ON / yes, and $0=$ false / OFF / no. Same idea, different costume.

2. A variable and its complement: $A$ and $\bar{A}$

Figure s01 — the bar as a mirror. The figure shows two situations stacked. Top row: a switch $A=1$ (glowing amber lamp) reflected across the dashed cyan "mirror" into $\bar{A}=0$ (dark lamp). Bottom row: $A=0$ (dark) reflects into $\bar{A}=1$ (glowing). The amber arrow is the act of flipping. Read it as: the bar takes whatever the lamp is doing and swaps it.

A literal is just "a variable, possibly barred": $A$ is a literal, $\bar{A}$ is a literal. That is the smallest brick.

3. The two operations: OR ($+$) and AND ($\cdot$)

The plus and dot signs in $(A+B+C)$ and $M_0\cdot M_2$ are not ordinary arithmetic. They are logic gates.

Figure s02 — OR is parallel, AND is series. Top half: two switches $A$ and $B$ on parallel wires feeding one lamp — either switch closing completes a path, so the lamp lights easily (shown ON). Bottom half: the same two switches on a single series wire — the current must pass through both, so if either is open the lamp is dark (shown OFF). This picture is the physical meaning of the two formulas that follow.

4. A row = one setting of all switches

5. Numbering the rows: reading binary

Each row also gets a number. We get it by reading the switch values as a binary number using the convention from the very top: $A$ (leftmost) is the high bit, $C$ (rightmost) the low bit.

Figure s03 — reading a row as a number. Three cyan-outlined buckets are labelled with the switch values $A=1$, $B=0$, $C=1$ and their place-values $4$, $2$, $1$. A bucket is filled amber only when its switch is $1$ (so the $A$ and $C$ buckets glow, the $B$ bucket stays empty). The amber tallies below add to the single number written at the bottom: $4+0+1=5$. This is precisely how a switch-snapshot becomes one index.

6. Minterm ($m_i$) vs Maxterm ($M_i$)

These are the atoms behind both forms — Minterms and maxterms is their home page.

7. The complement of a function: $\bar{F}$

8. The big product symbol $\prod$

Prerequisite map

Equipment checklist

Cover the right side and answer each. If any stumps you, re-read its section above before opening the POS note.