💡 Minterms and Maxterms

Any system has inputs and outputs, and can have outputs that represent ON or OFF states

Our goal is to derive Boolean expression(s) that fully reflects the relationship between the inputs of a system to its active states——ON or OFF

☯️ ON and OFF States

A function’s output is either ON or OFF

Defining a System We can represent the system through

All input combinations that turn it ON
Or all combinations that turn it OFF

Defining one automatically defines the other—this is the Principle of Duality

We only need to choose one side to work with.

🔌Importance of Active States

When designing systems that has active states——ON and OFF——we have to determine which state we want to focus on that will represent the behaviour of the system As the same system cannot be represented by both ON and OFF at the same time

They must be represented separately

For us, we would prefer to think of existence as a more meaningful and important state than non-existence

So we will focus on which what represents existence——ON——to represent systems with active states

💈Strictness of Active States

We also prefer to have the state that we care about——ON——to have a stricter condition

This is such that we can control exactly which combinations of inputs in our systems will definitely lead to its activation

In the same vein, OFF will be treated as the more lenient condition

📐Activation in Binary

The binary numbers picked for activation is also reflected as what we prefer to be active, where 1 represents High Electrical Signal and 0 represents Low Electrical Signal

The convention used for representing activation states is also called Active High

🧱 Representation of Each Input Combination by Active State

All the states of a system can be listed out in what we call a Truth Table

A	B	C	F
0	0	0	1
0	0	1	0
0	1	0	1
0	1	1	1
1	0	0	0
1	0	1	1
1	1	0	0
1	1	1	1

Each row of such a table contains a unique input combination of the system——A,B,C in this case——that relates to its corresponding output——either 1 representing ON or 0 representing OFF

Strictness of Combinations Thus each ON state of the system can be represented by strict combination of all its corresponding inputs

And each OFF state of the system can be represented by lenient combination of all its corresponding inputs

These concepts will be coined and explained in further detail in the below sub-sections

🔹 Minterm

The strict combination of all the corresponding inputs that causes the output to have an ON state, is called a Minterm

Etymology It is also the minimum combination-of-inputs condition for the system to be considered ON

Thus, the term representing such a combination of inputs is called the Minimum Term

Or the Minterm

Strictness is represented by the AND operator, so its inputs will be AND-ed to represent the ON state

Since AND requires all its inputs to be True for the AND expression to be True, the inputs that are 0 in that row are represented through the complement of that input variable

For $A=1$, $B=0$, $C=1$, the minterm is:

$$ A \cdot B' \cdot C $$

This is only true when A is True, B is False, and C is True

🔸 Maxterm

The lenient combination of all the corresponding inputs that causes the output to have an OFF state, is called a Maxterm

Etymology It is also the maximum combination-of-inputs condition for the system to be considered OFF

Thus, the term representing such a combination of inputs is called the Maximum Term

Or the Maxterm

Lenience is represented by the OR operator, so its inputs will be OR-ed to represent the OFF state

Since OR requires none of its inputs to be True for the OR expression to be False, the inputs that are 1 in that row are represented through the complement of that input variable

For $A=1$, $B=0$, $C=1$, the maxterm is:

$$ A' + B + C' $$

This is only false when A is True, B is False, and C is True

🧮 Representation of the System by Active State

Now that we have an expression representing each ON state of the system through the strict combination of all its corresponding inputs——the minterm

And also an expression representing each OFF state of the system through the lenient combination of all its corresponding inputs——the maxterm

We can represent the system itself through the combinations of all of either the expressions representing the ON states of the system——minterms

the expressions representing the OFF states of the system——maxterms

The System Represented Through ON States Since each separate condition for the ON state is strict, we only need either of them for the whole system to be ON

Thus, the representation of the system through all the ON states is lenient

ℹ️

The System Represented Through OFF States Since each separate condition for the OFF state is lenient, we must exclude all of them for the whole system to be OFF

Thus, the representation of the system through all the OFF states is strict

These concepts will be coined and explained in further detail in the below sub-sections

➕ Sum of Minterms

The system represented by ON state, through the lenient combination of minterms, is called the Sum of Minterms

It is named so because a lenient combination is what OR represents, which written out in Boolean Algebraic Form uses the + sign to represent OR

Take the same Truth Table shown earlier

A	B	C	F
0	0	0	1
0	0	1	0
0	1	0	1
0	1	1	1
1	0	0	0
1	0	1	1
1	1	0	0
1	1	1	1

The Sum of Minterms will be represented and calculated like this

$$ \begin{align} F(A, B, C) &= \sum m(1,3,4,6,8) \\ &= m_1 + m_3 + m_4 + m_6 + m_8 \\ &= A'B'C' + A'BC' + A'BC + AB'C + ABC \end{align} $$

This is read as:

The general function, F, with input variables, A,B,C, is the sum, Σ, of minterms, m, in rows 1,3,4,6,8

This full exhaustive form is known as the Canonical Form, as it contains all the terms that contain all the input variables

This expression is thus more aptly known as the Sum of Products, or SOP for short

Take note that Sum here is NOT the arithmetic sum used in Maths, but a borrowed symbol to represent OR in Logic

The structural similarity between logical OR and arithmetic sum——such as associativity, commutativity, and distributivity——is why the symbol was borrowed

Any further simplification of Sum of Products can be done through an optimised method, leveraging visualisation, called Karnaugh Maps

✖️ Product of Maxterms

The system represented by OFF state, through the strict combination of maxterms, is called the Product of Maxterms

It is named so because a strict combination is what AND represents, which written out in Boolean Algebraic Form uses the ⋅ sign to represent AND

Take the same Truth Table shown earlier

A	B	C	F
0	0	0	1
0	0	1	0
0	1	0	1
0	1	1	1
1	0	0	0
1	0	1	1
1	1	0	0
1	1	1	1

The Product of Maxterms will be represented and calculated like this

$$ \begin{align} F(A, B, C) &= \prod M(2,5,7) \\ &= M_2 \cdot M_5 \cdot M_7 \\ &= (A + B + C') \cdot (A + B' + C) \cdot (A' + B + C) \end{align} $$

This is read as:

The general function, F, with input variables, A,B,C, is the product, Π, of maxterms, M, in rows 2,5,7

This full exhaustive form is known as the Canonical Form, as it contains all the terms that contain all the input variables

This expression is thus more aptly known as the Product of Sums, or POS for short

Take note that Product here is NOT the arithmetic product used in Maths, but a borrowed symbol to represent AND in Logic

The structural similarity between logical AND and arithmetic product——such as associativity, commutativity, and distributivity——is why the symbol was borrowed

Any further simplification of Product of Sums can be done through an optimised method, leveraging visualisation, called Karnaugh Maps

⚖️ De Morgan Laws

Both SOP and POS represent the same function—just from opposite perspectives.

De Morgan’s Laws De Morgan’s Laws allow us to interchange AND/OR operations transform between SOP and POS

$$ (A \cdot B)' = A' + B' $$$$ (A + B) = A' \cdot B' $$

📋 Summary Table

Logic Basis	Concept	Primary Goal	Operator for Terms	Operator to Combine Terms
Active High Logic	Sum of Minterms OR Sum of Products or SOP	Describe all “ON” states (1s)	AND (strict)	OR (inclusive)
Active High Logic	Product of Maxterms OR Product of Sums or POS	Describe all “OFF” states (0s)	OR (less strict)	AND (strict)

Last updated on 2025-11-08

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