Related Concepts: 04 Mathematical Conversions & Expansion of SOP and POS | 01 Boolean Algebra Foundations & Duality Principle | 02 Boolean Algebra Significance & Circuit Minimization

3.03 SOP, POS, Canonical & Standard Forms

Introduction to Boolean Logic Forms

Any Boolean function can be expressed in two standard algebraic structures: Sum of Products (SOP) or Product of Sums (POS).

graph TD
    BF[Boolean Expressions] --> SOP[Sum of Products SOP]
    BF --> POS[Product of Sums POS]
    
    SOP -->|OR-ing AND terms| CSOP[Canonical SOP / Sum of Minterms Σm]
    SOP -->|Simplified| SSOP[Standard SOP e.g. AB + BC]
    
    POS -->|AND-ing OR terms| CPOS[Canonical POS / Product of Maxterms ΠM]
    POS -->|Simplified| SPOS[Standard POS e.g. A+B B+C]

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1. Canonical Forms vs. Standard Forms

A. The Canonical Form (Strictly Unique)

Definition: Canonical Form

A Boolean expression is in Canonical Form if every term contains all domain variables of the function exactly once, in either complemented or uncomplemented form. Canonical forms are strictly unique.

• Minterm ($m_j$): An AND product term containing all $n$ domain variables. Evaluates to 1 for exactly one input state. Used in Canonical SOP ($\Sigma m$).
• Maxterm ($M_j$): An OR sum term containing all $n$ domain variables. Evaluates to 0 for exactly one input state. Used in Canonical POS ($\Pi M$).

B. The Standard Form (Simplified / Non-Unique)

Definition: Standard Form

A Boolean expression is in Standard Form if it follows SOP or POS structure, but terms are simplified and do not need to contain all domain variables. Standard forms are non-unique.

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2. Comparison Table: SOP vs. POS

Parameter	Sum of Products (SOP)	Product of Sums (POS)
Basic Structure	Formed by OR-ing multiple AND product terms.	Formed by AND-ing multiple OR sum terms.
Truth Table Source	Derived from rows where output $F=1$.	Derived from rows where output $F=0$.
Canonical Component	Sum of Minterms ($\Sigma m$).	Product of Maxterms ($\Pi M$).
Native Gate Realization	2-level AND-OR or NAND-NAND logic.	2-level OR-AND or NOR-NOR logic.

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3. PYQ Master Problem: POS Canonical Extraction

PYQ Master Problem (2016): POS Canonical Extraction

Question: Given the function $f(A,B,C)=(A+B+C^{\prime })(A+B^{\prime }+C^{\prime })(A^{\prime }+B+C^{\prime })(A^{\prime }+B^{\prime }+C^{\prime })$, construct the truth table and express the function in maxterm and minterm forms.

Step 1: Identify the Maxterms The function is given in canonical Product of Sums (POS) form. A maxterm evaluates to 0 when its uncomplemented variables are 0 and its complemented variables are 1:

• $(A+B+C^{\prime })\to 001_2\to {\mathbf{M}}_1$
• $(A+B^{\prime }+C^{\prime })\to 011_2\to {\mathbf{M}}_3$
• $(A^{\prime }+B+C^{\prime })\to 101_2\to {\mathbf{M}}_5$
• $(A^{\prime }+B^{\prime }+C^{\prime })\to 111_2\to {\mathbf{M}}_7$

Maxterm Form: $\mathbf{f}(\mathbf{A},\mathbf{B},\mathbf{C})=\mathbf{\Pi M}(1,3,5,7)$

Step 2: Identify the Minterms Minterms are the remaining decimal states (0 through 7) not present in the maxterm list: Minterm Form: $\mathbf{f}(\mathbf{A},\mathbf{B},\mathbf{C})=\mathbf{\Sigma m}(0,2,4,6)$

Step 3: Construct the Truth Table

A B C	f(A,B,C)
0 0 0	1
0 0 1	0
0 1 0	1
0 1 1	0
1 0 0	1
1 0 1	0
1 1 0	1
1 1 1	0

(Notice $f(A,B,C)$ simplifies to $f=C^{\prime }$). $■$

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Past Year Questions (PYQs)

• [PYQ 2015, 2016, 2019, 2021, 2022, 2024]: Define Canonical form vs. Standard form.
• [PYQ 2021, 2022]: Distinguish between canonical form and standard form of a Boolean function. (08 to 10 Marks)
• [PYQ 2016]: POS Canonical Extraction, truth table construction, minterm/maxterm forms. (10 Marks)