💡 The Sixteen Faces of Logic: All Possible Boolean Functions with Two Inputs 🔗

Boolean algebra, named after mathematician George Boole, is the bedrock of all digital circuitry. It deals with variables that can only have two states: True (1) or False (0). When we consider two input variables, A and B, there are only four possible combinations of inputs: (0, 0), (0, 1), (1, 0), and (1, 1).

Since each of these four input combinations can result in either a 0 or a 1 as an output, the total number of unique Boolean functions possible is $2^4 = 16$ .

This article explores these 16 fundamental logical operations, categorized by their complexity and importance in digital design.

📑 Table of Contents 🔗

🔢 The Four Input Combinations 🔗

To understand the functions, we must first define the input possibilities.

A	B
0	0
0	1
1	0
1	1

🚀 Categories of the 16 Functions 🔗

The 16 functions can be grouped into several categories based on their output patterns, from the simplest constants to the most complex exclusive gates. Let $F$ represent the function’s output.

⚪ 1. Constant Functions (2 Functions) 🔗

These functions ignore the inputs entirely and always produce the same output.

Function Name	Expression	A=0, B=0	A=0, B=1	A=1, B=0	A=1, B=1	Description
Zero (FALSE)	$F_0 = 0$	0	0	0	0	Always outputs 0.
One (TRUE)	$F_{15} = 1$	1	1	1	1	Always outputs 1.

🔄 2. Simple Transfer/Inversion Functions (4 Functions) 🔗

These functions depend on only one of the inputs, either directly or inverted.

Function Name	Expression	A=0, B=0	A=0, B=1	A=1, B=0	A=1, B=1	Description
Transfer A	$F_3 = A$	0	0	1	1	Output is simply A.
Transfer B	$F_5 = B$	0	1	0	1	Output is simply B.
NOT A	$F_{12} = \bar{A}$	1	1	0	0	Output is the inverse of A.
NOT B	$F_{10} = \bar{B}$	1	0	1	0	Output is the inverse of B.

⚙️ 3. Basic Binary Operations (4 Functions) 🔗

These are the primary logic gates used universally in digital systems.

Function Name	Expression	A=0, B=0	A=0, B=1	A=1, B=0	A=1, B=1	Common Gate
AND	$F_8 = A \cdot B$	0	0	0	1	A 1 only if A AND B are 1.
OR	$F_{14} = A + B$	0	1	1	1	A 1 if A OR B (or both) is 1.
NAND	$F_7 = \overline{A \cdot B}$	1	1	1	0	The NOT-AND of A and B. (Universal Gate)
NOR	$F_1 = \overline{A + B}$	1	0	0	0	The NOT-OR of A and B. (Universal Gate)

⚖️ 4. Exclusive Operations (2 Functions) 🔗

These functions are critical for arithmetic circuits (like adders) and comparing inputs.

Function Name	Expression	A=0, B=0	A=0, B=1	A=1, B=0	A=1, B=1	Description
XOR (Exclusive-OR)	$F_6 = A \oplus B$	0	1	1	0	A 1 if A and B are different.
XNOR (Equivalence)	$F_9 = \overline{A \oplus B}$	1	0	0	1	A 1 if A and B are the same.

➡️ 5. Conditional and Inverse Conditional Functions (4 Functions) 🔗

These functions are less common in general digital logic but play a role in advanced logic design and certain formal systems.

Function Name	Expression	A=0, B=0	A=0, B=1	A=1, B=0	A=1, B=1	Description
A Implies B (Material Implication)	$F_{13} = \bar{A} + B$	1	1	0	1	False only if A is 1 and B is 0.
B Implies A	$F_{11} = A + \bar{B}$	1	0	1	1	False only if B is 1 and A is 0.
A NOT-Implies B	$F_4 = A \cdot \bar{B}$	0	0	1	0	True only if A is 1 and B is 0.
B NOT-Implies A	$F_2 = \bar{A} \cdot B$	0	1	0	0	True only if A is 0 and B is 1.

🔑 The Power of Universality 🔗

The most fascinating aspect of these 16 functions is that two are individually capable to construct all the others: the NAND gate and the NOR gate.

NAND ( $F_7$ ) can be wired to perform AND, OR, NOT, and XOR operations.
NOR ( $F_1$ ) can also be wired to perform AND, OR, NOT, and XOR operations.

Because of this property, NAND and NOR gates are known as Universal Gates. A computer can be built entirely using only one type of gate, significantly simplifying the manufacturing process of integrated circuits.

📝 Summary Table of All 16 Functions 🔗

This table consolidates all 16 functions based on the output for the four input combinations.

F#	(0,0)	(0,1)	(1,0)	(1,1)	Function Name	Expression	Common Gate
$F_0$	0	0	0	0	Zero	$0$	(Constant)
$F_1$	1	0	0	0	NOR	$\overline{A + B}$	NOR
$F_2$	0	1	0	0	B NOT-Implies A	$\bar{A} \cdot B$
$F_3$	0	0	1	1	Transfer A	$A$
$F_4$	0	0	1	0	A NOT-Implies B	$A \cdot \bar{B}$
$F_5$	0	1	0	1	Transfer B	$B$
$F_6$	0	1	1	0	XOR	$A \oplus B$	XOR
$F_7$	1	1	1	0	NAND	$\overline{A \cdot B}$	NAND
$F_8$	0	0	0	1	AND	$A \cdot B$	AND
$F_9$	1	0	0	1	XNOR	$\overline{A \oplus B}$	XNOR
$F_{10}$	1	0	1	0	NOT B	$\bar{B}$	NOT
$F_{11}$	1	0	1	1	B Implies A	$A + \bar{B}$
$F_{12}$	1	1	0	0	NOT A	$\bar{A}$	NOT
$F_{13}$	1	1	0	1	A Implies B	$\bar{A} + B$
$F_{14}$	0	1	1	1	OR	$A + B$	OR
$F_{15}$	1	1	1	1	One	$1$	(Constant)

These 16 logical functions, derived from two simple binary inputs, form the complete vocabulary for all computational processes, demonstrating the incredible complexity that can be built from the simplest of foundations.

📚 Download My Free Ebook 🔗

Get the curated offline edition of Kaue's Way of Life—a practical field guide to optimizing 🍏 Health, 💰 Wealth, ❤️ Love, and 🧘 Meaning.

📬 Join My Mailing List 🔗

Get notified when I publish new essays, book summaries, and practical guides on living well. No spam—unsubscribe anytime.

💬 Comments & Discussion 🔗

DISCLAIMER: The content on this site is for informational and educational purposes only and does not constitute medical, psychological, financial, investment, tax, or legal advice. The ideas, curation, and opinions presented here are my own and do not represent those of my employer. Book, lecture, and article summaries are independent educational commentary and remain the intellectual property of their respective authors. AI tools may be used to assist with editing, formatting, and synthesis; while every effort is made to ensure accuracy, errors may occur. PRIVACY: This site uses third-party services for cookieless analytics (Google Analytics), optional newsletter subscriptions (Mailchimp), comments (Disqus), live chat (Tawk.to), and embedded media (YouTube), which may process visitor IP addresses and cookies in accordance with their respective privacy policies.