Sum of products (SOP) form

WHAT is SOP?

• Sum = OR ($+$), because OR behaves like addition in Boolean algebra.
• Product = AND ($\cdot$), because AND behaves like multiplication.

WHY does SOP work? (first principles)

A minterm is a row-detector. Consider variables $A,B,C$ and the row $A=1, B=0, C=1$.

Build the product $A\bar{B}C$:

• It equals 1 only when $A=1$ (so $A=1$ ✓), $\bar B=1$ (so $B=0$ ✓), $C=1$ ✓.
• For any other combination at least one literal is 0, so the AND is 0.

So each minterm outputs 1 for exactly one input row and 0 everywhere else.

Now OR together the minterms of all rows where $F=1$. Because OR gives 1 if any term is 1:

• On a "$F=1$" row → its own minterm fires → output 1. ✓
• On a "$F=0$" row → no minterm fires → output 0. ✓

This reproduces the truth table exactly. That is the proof that every Boolean function has an SOP form.

HOW to construct SOP — step by step

Method: for each row with output 1, write the minterm (0 → complement, 1 → true), then OR them.

Common Mistakes (Steel-manned)

Active Recall

Flashcards

Connections

• Truth Tables — SOP is read directly off the 1-rows.
• Minterms and Maxterms — SOP built from minterms; POS from maxterms.
• Product of Sums (POS) form — the dual method.
• Karnaugh Maps — visual tool to simplify SOP.
• Boolean Algebra Laws — distribution, absorption, complement laws used to simplify.
• Logic Gates (AND, OR, NOT) — each product = AND gate, joined by an OR gate.

Concept Map

Hinglish (regional understanding)