---
title: "SOP/POS forms and canonical form of A+BC+ABC"
subject: "Digital Logic"
chapter: 3
key: "semester_3/digital_logic:sop-pos-canonical-form"
source: https://www.bctengineer.com/q/semester_3/digital_logic/sop-pos-canonical-form
site: BCT Engineering
---

# SOP/POS forms and canonical form of A+BC+ABC

Digital Logic · Chapter 3: Combinational Logic Circuits

- Asked in: 2078 Bhadra, 2+2 marks
- Video: [SOP and POS Form : Non Canonical to Canonical Form Conversion of Boolean Expression](https://www.youtube.com/watch?v=yFEM6uq4jbY) — ALL ABOUT ELECTRONICS
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/digital_logic/sop-pos-canonical-form
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-3

## Question

Define SOP and POS form and convert F = A + BC + ABC into its canonical form.

## Short answer

SOP is OR of AND terms; POS is AND of OR terms. Canonical SOP: F = Σm(3,4,5,6,7). Canonical POS: F = ΠM(0,1,2).

## Answer

#### SOP and POS forms

- **Sum of Products (SOP):** a logical sum (OR) of product (AND) terms, e.g. $F = AB + \overline{A}C$. It is built as an AND-OR (or NAND-NAND) circuit. If every term has all the variables, it is the canonical SOP (sum of minterms).
- **Product of Sums (POS):** a logical product (AND) of sum (OR) terms, e.g. $F = (A+B)(\overline{A}+C)$. It is built as an OR-AND (or NOR-NOR) circuit. If every term has all the variables, it is the canonical POS (product of maxterms).

#### Canonical form of $F = A + BC + ABC$

Expand each term to contain A, B and C:
$$
\begin{aligned} A &= A(B+\overline{B})(C+\overline{C}) = ABC + AB\overline{C} + A\overline{B}C + A\overline{B}\,\overline{C} \\ BC &= BC(A+\overline{A}) = ABC + \overline{A}BC \\ ABC &= ABC \end{aligned}
$$

Add them and keep each term once ($X+X=X$):
$$
F = \overline{A}BC + A\overline{B}\,\overline{C} + A\overline{B}C + AB\overline{C} + ABC
$$

| Term | ABC | Minterm |
|---|---|---|
| $\overline{A}BC$ | 011 | $m_3$ |
| $A\overline{B}\,\overline{C}$ | 100 | $m_4$ |
| $A\overline{B}C$ | 101 | $m_5$ |
| $AB\overline{C}$ | 110 | $m_6$ |
| $ABC$ | 111 | $m_7$ |

**Canonical SOP:**
$$
F = \Sigma m(3,4,5,6,7)
$$

**Canonical POS:** the missing numbers 0, 1, 2 are the maxterms:
$$
F = \Pi M(0,1,2) = (A+B+C)(A+B+\overline{C})(A+\overline{B}+C)
$$

(The given function itself simplifies to $F = A + BC$, since $ABC$ is absorbed by $A$.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which statement defines SOP?
   A. An AND of OR terms
   B. An OR of AND terms
   C. An OR of OR terms
   D. An AND of NOT terms
   Answer: B. Sum of products is the logical sum (OR) of product (AND) terms, AND-OR or NAND-NAND. (Key point: SOP definition)

2. Which circuit type does a POS expression naturally map to?
   A. AND-OR
   B. OR-AND (or NOR-NOR)
   C. NAND-NAND
   D. XOR-AND
   Answer: B. POS is an AND of OR terms, so OR-AND, or NOR-NOR using universal gates. (Key point: POS circuit)

3. What is the canonical SOP?
   A. Σm(3,4,5,6,7)
   B. Σm(0,1,2)
   C. Σm(4,5,6,7)
   D. Σm(1,3,5,7)
   Answer: A. A gives 4-7, BC gives 3 and 7, and ABC gives 7. (Key point: Canonical SOP of A + BC + ABC)

4. What is the canonical POS of the same function?
   A. ΠM(3,4,5,6,7)
   B. ΠM(0,1,2)
   C. ΠM(7)
   D. ΠM(4,5,6)
   Answer: B. Maxterms are the missing numbers 0,1,2. (Key point: Canonical POS)

## Related in Digital Logic

- [K-map Σ(2,3,6,7,9,10,11)+d in SOP and POS](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-sop-pos-2367) — Digital Logic, chapter 3
- [Octal-to-binary encoder; A+B'C canonical form](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Aoctal-binary-encoder-canonical) — Digital Logic, chapter 4
- [Canonical form of A(B'+C)+BD; minterm/maxterm](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Acanonical-form-minterm-maxterm) — Digital Logic, chapter 3
- [Minimal SOP Σm(3,4,6,8,10,15)+d, NOR-only](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-minimal-sop-nor) — Digital Logic, chapter 3
- [Minimize 9-term SOP and implement in NAND](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Aminimize-nine-term-nand) — Digital Logic, chapter 3

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