---
title: "Canonical form of A(B'+C)+BD; minterm/maxterm"
subject: "Digital Logic"
chapter: 3
key: "semester_3/digital_logic:canonical-form-minterm-maxterm"
source: https://www.bctengineer.com/q/semester_3/digital_logic/canonical-form-minterm-maxterm
site: BCT Engineering
---

# Canonical form of A(B'+C)+BD; minterm/maxterm

Digital Logic · Chapter 3: Combinational Logic Circuits

- Asked in: 2078 Kartik, 3+2 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/digital_logic/canonical-form-minterm-maxterm
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-3

## Question

Given function F = A(B'+C)+BD, change into its canonical form. Define max term and min term.

## Short answer

Canonical SOP: F = Σm(5,7,8,9,10,11,13,14,15); POS: F = ΠM(0,1,2,3,4,6,12). A minterm is a product of all variables (true/complemented) that equals 1 for exactly one input combination. A maxterm is a sum of all variables that equals 0 for exactly one input combination.

## Answer

#### Canonical form of $F = A(\overline{B}+C) + BD$

**Step 1: write F in SOP form**
$$
F = A\overline{B} + AC + BD
$$

**Step 2: expand each term to include all four variables A, B, C, D**
$$
\begin{aligned} A\overline{B} &= A\overline{B}(C+\overline{C})(D+\overline{D}) = A\overline{B}CD + A\overline{B}C\overline{D} + A\overline{B}\,\overline{C}D + A\overline{B}\,\overline{C}\,\overline{D} \\ &= m_{11}+m_{10}+m_9+m_8 \\ AC &= AC(B+\overline{B})(D+\overline{D}) = ABCD + ABC\overline{D} + A\overline{B}CD + A\overline{B}C\overline{D} \\ &= m_{15}+m_{14}+m_{11}+m_{10} \\ BD &= BD(A+\overline{A})(C+\overline{C}) = ABCD + AB\overline{C}D + \overline{A}BCD + \overline{A}B\overline{C}D \\ &= m_{15}+m_{13}+m_7+m_5 \end{aligned}
$$

**Step 3: combine, keeping each minterm once**

Canonical SOP:
$$
F = \Sigma m(5,7,8,9,10,11,13,14,15)
$$
$$
\begin{aligned} F = {} & \overline{A}B\overline{C}D + \overline{A}BCD + A\overline{B}\,\overline{C}\,\overline{D} + A\overline{B}\,\overline{C}D + A\overline{B}C\overline{D} \\ & + A\overline{B}CD + AB\overline{C}D + ABC\overline{D} + ABCD \end{aligned}
$$

Canonical POS (the remaining numbers):
$$
F = \Pi M(0,1,2,3,4,6,12)
$$
$$
\begin{aligned} F = {} & (A+B+C+D)(A+B+C+\overline{D})(A+B+\overline{C}+D)(A+B+\overline{C}+\overline{D}) \\ & (A+\overline{B}+C+D)(A+\overline{B}+\overline{C}+D)(\overline{A}+\overline{B}+C+D) \end{aligned}
$$

#### Minterm and maxterm

- **Minterm:** a product term containing all variables once (true or complemented); it is 1 for exactly one input combination. Example: $m_5 = \overline{A}B\overline{C}D$ is 1 only for ABCD = 0101.
- **Maxterm:** a sum term containing all variables once (true or complemented); it is 0 for exactly one input combination. Example: $M_0 = A+B+C+D$ is 0 only for ABCD = 0000.
- $M_i = \overline{m_i}$; canonical SOP is a sum of minterms and canonical POS is a product of maxterms.

## Check yourself

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

1. What is the first step when converting F = A(B'+C)+BD to canonical form?
   A. Write in SOP form: AB' + AC + BD
   B. Complement the function
   C. Make a truth table of only A
   D. Write in POS form
   Answer: A. Expanding the brackets gives an SOP that can be expanded with missing variables. (Key point: First step)

2. AB' expands in four variables to which minterms?
   A. m8,m9,m10,m11
   B. m4,m5,m6,m7
   C. m10,m11,m14,m15
   D. m0,m1,m2,m3
   Answer: A. AB' = 10xx, giving 1000 to 1011 = 8,9,10,11. (Key point: Expanding AB')

3. What is the canonical SOP?
   A. Σm(5,7,8,9,10,11,13,14,15)
   B. Σm(8,9,10,11)
   C. Σm(0,1,2,3,4,6,12)
   D. Σm(5,7,13,15)
   Answer: A. Combine AB' (8-11), AC (10,11,14,15) and BD (5,7,13,15). (Key point: Canonical SOP)

4. What is the canonical POS?
   A. ΠM(5,7,8,9,10,11,13,14,15)
   B. ΠM(0,1,2,3,4,6,12)
   C. ΠM(8,9,10,11)
   D. ΠM(0,15)
   Answer: B. The maxterms are the numbers missing from the SOP list. (Key point: Canonical POS)

## Related in Digital Logic

- [A+B'C to standard minterm form](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Astandard-minterm-a-plus-bc) — Digital Logic, chapter 3
- [SOP/POS forms and canonical form of A+BC+ABC](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Asop-pos-canonical-form) — 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
- [Maxterm with example](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Amaxterm-explain) — Digital Logic, chapter 3
- [K-map Σ(0,1,4,8,10,11,12)+d, standard minterm](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-standard-minterm-01248) — Digital Logic, chapter 3
- [Minterms/maxterms; K-map Σm(2,4,5,13,14)+d, universal gates](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Aminterm-maxterm-kmap-universal) — Digital Logic, chapter 3

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