KAHIBARO
Table of Contents
Logical Variables and Operations
Boolean algebra is the mathematics of logic. In digital electronics, it is used to describe how circuits make decisions using only two states, usually written as 0 and 1. A Boolean variable can have only one of these two values. In practical circuits, 0 often represents a low voltage and 1 a high voltage, but Boolean algebra itself is about logic, not about the physical voltage levels.
The main purpose of Boolean algebra is to express logical relationships in a clear symbolic form. A circuit may take several binary inputs and produce a binary output. Boolean expressions let us describe that output exactly.
A Boolean variable is often written with a letter such as $A$, $B$, or $X$. The three most basic operations are AND, OR, and NOT. These operations are the foundation of digital logic.
The Basic Operations
The AND operation gives an output of 1 only when all required inputs are 1. It is written as $A \cdot B$, or sometimes simply $AB$.
The OR operation gives an output of 1 when at least one input is 1. It is written as $A + B$.
The NOT operation reverses the value of a variable. If $A = 1$, then NOT $A$ is 0. If $A = 0$, then NOT $A$ is 1. It is written as $\overline{A}$, $A'$, or $\neg A$. In beginner electronics, $\overline{A}$ is very common.
The truth tables below show these operations.
| $A$ | $B$ | $A \cdot B$ |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
| $A$ | $B$ | $A + B$ |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
| $A$ | $\overline{A}$ | |
| 0 | 1 | |
| 1 | 0 |
Important notation:
$A \cdot B$ means AND
$A + B$ means OR
$\overline{A}$ means NOT $A$
These symbols do not mean ordinary arithmetic, even though they look similar.
Combining Operations
Boolean expressions can combine several operations. For example,
$$
Y = A \cdot \overline{B}
$$
means that the output $Y$ is 1 only when $A = 1$ and $B = 0$.
Another example is
$$
Y = A + B \cdot C
$$
This means that $B \cdot C$ is evaluated first, then ORed with $A$. As in ordinary algebra, parentheses help remove ambiguity. For instance,
$$
Y = (A + B)\cdot C
$$
is not the same as
$$
Y = A + B\cdot C
$$
In the first case, the OR happens first. In the second case, the AND happens first.
Fundamental Laws of Boolean Algebra
Boolean algebra has its own set of laws. These laws let us simplify expressions, which is very useful when designing circuits.
Identity Laws
The identity laws show how 0 and 1 behave in Boolean expressions.
$$
A + 0 = A
$$
$$
A \cdot 1 = A
$$
Adding 0 with OR changes nothing. Multiplying by 1 with AND changes nothing.
Null Laws
The null laws show the dominating effect of 1 in OR and 0 in AND.
$$
A + 1 = 1
$$
$$
A \cdot 0 = 0
$$
If one input of an OR is already 1, the output must be 1. If one input of an AND is 0, the output must be 0.
Idempotent Laws
Repeating the same variable does not change the result.
$$
A + A = A
$$
$$
A \cdot A = A
$$
Complement Laws
A variable combined with its complement gives a fixed result.
$$
A + \overline{A} = 1
$$
$$
A \cdot \overline{A} = 0
$$
Double Negation Law
Taking the complement twice returns the original variable.
$$
\overline{\overline{A}} = A
$$
Commutative Laws
The order of variables does not matter for AND and OR.
$$
A + B = B + A
$$
$$
A \cdot B = B \cdot A
$$
Associative Laws
Grouping does not matter when the same operation repeats.
$$
(A + B) + C = A + (B + C)
$$
$$
(A \cdot B)\cdot C = A \cdot (B \cdot C)
$$
Distributive Laws
These laws connect AND and OR.
$$
A \cdot (B + C) = A\cdot B + A\cdot C
$$
$$
A + B\cdot C = (A + B)(A + C)
$$
The second form may look unusual because it is different from ordinary algebra, but it is valid in Boolean algebra.
Core Boolean laws to remember:
$A + 0 = A$, $A \cdot 1 = A$
$A + 1 = 1$, $A \cdot 0 = 0$
$A + \overline{A} = 1$, $A \cdot \overline{A} = 0$
$A \cdot (B + C) = A\cdot B + A\cdot C$
$A + B\cdot C = (A + B)(A + C)$
Absorption Laws
Absorption laws are especially useful for simplification.
$$
A + A\cdot B = A
$$
$$
A \cdot (A + B) = A
$$
These say that one term can absorb a more complicated one.
For example,
$$
X = A + A\cdot B
$$
can be reduced immediately to
$$
X = A
$$
because if $A$ is 1, the output is already 1, and if $A$ is 0, then $A\cdot B$ is also 0.
De Morgan's Theorems
De Morgan's theorems are among the most important results in Boolean algebra. They tell us how complements interact with AND and OR.
$$
\overline{A \cdot B} = \overline{A} + \overline{B}
$$
$$
\overline{A + B} = \overline{A}\cdot \overline{B}
$$
These rules extend to more than two variables.
$$
\overline{A \cdot B \cdot C} = \overline{A} + \overline{B} + \overline{C}
$$
$$
\overline{A + B + C} = \overline{A}\cdot \overline{B}\cdot \overline{C}
$$
De Morgan's theorems are very useful when transforming one circuit form into another.
De Morgan's theorems:
$\overline{A \cdot B} = \overline{A} + \overline{B}$
$\overline{A + B} = \overline{A}\cdot \overline{B}$
When a complement moves across parentheses, AND changes to OR, and OR changes to AND. Every variable inside is also complemented.
Example of Simplification
Suppose we want to simplify
$$
Y = A\cdot B + A\cdot \overline{B}
$$
We can factor out $A$:
$$
Y = A(B + \overline{B})
$$
Using the complement law,
$$
B + \overline{B} = 1
$$
so
$$
Y = A\cdot 1 = A
$$
This means the whole expression simplifies to
$$
Y = A
$$
A more compact expression usually means a simpler circuit.
Another Example
Simplify
$$
Y = (A + B)(A + \overline{B})
$$
Using the distributive law in reverse,
$$
(A + B)(A + \overline{B}) = A + B\cdot \overline{B}
$$
Since
$$
B\cdot \overline{B} = 0
$$
we get
$$
Y = A + 0 = A
$$
Again, the result is simply
$$
Y = A
$$
Truth Table Verification
One way to check a Boolean expression is to build its truth table. For the expression
$$
Y = A\cdot B + A\cdot \overline{B}
$$
we can evaluate every possible input pair.
| $A$ | $B$ | $\overline{B}$ | $A\cdot B$ | $A\cdot \overline{B}$ | $Y$ |
|---|---|---|---|---|---|
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 | 0 | 1 |
The output column matches $A$, confirming that
$$
A\cdot B + A\cdot \overline{B} = A
$$
Boolean Algebra and Circuit Meaning
Boolean algebra is directly connected to logic circuits. A Boolean expression tells us how signals combine. If an expression is simplified, the corresponding circuit can often be built with fewer gates. Fewer gates usually means lower cost, less power use, and sometimes faster operation.
For example, the unsimplified expression
$$
Y = A\cdot B + A\cdot \overline{B}
$$
needs an inverter, two AND operations, and one OR operation. After simplification to
$$
Y = A
$$
the circuit no longer depends on $B$ at all.
Common Algebraic Patterns
Some expression forms appear often in digital logic.
| Expression | Simplified form | Reason |
|---|---|---|
| $A + A\cdot B$ | $A$ | Absorption |
| $A(A + B)$ | $A$ | Absorption |
| $A\cdot B + A\cdot \overline{B}$ | $A$ | Complement law |
| $(A + B)(A + \overline{B})$ | $A$ | Complement law |
| $\overline{\overline{A}}$ | $A$ | Double negation |
| $\overline{A\cdot B}$ | $\overline{A} + \overline{B}$ | De Morgan |
| $\overline{A + B}$ | $\overline{A}\cdot \overline{B}$ | De Morgan |
Recognizing these patterns makes simplification much easier.
Difference from Ordinary Algebra
Boolean algebra uses symbols that resemble ordinary algebra, but the rules are not identical. In ordinary algebra, $A + A = 2A$. In Boolean algebra,
$$
A + A = A
$$
because ORing a value with itself changes nothing. Also, in ordinary algebra, $A^2$ is not usually equal to $A$, but in Boolean algebra,
$$
A\cdot A = A
$$
because ANDing a value with itself changes nothing.
This difference is important. Boolean algebra works with logical truth values, not with arbitrary numbers.
Why Simplification Matters
A Boolean expression can often be written in several equivalent forms. All of them produce the same truth table, but some are much easier to implement. Simplification is therefore a practical tool, not just a mathematical exercise.
If two expressions are equivalent, they give the same output for every possible input combination. Boolean algebra gives us a way to prove that equivalence without having to build the entire circuit physically.
Two Boolean expressions are equivalent if they produce the same output for every possible input combination.
Simplifying a Boolean expression does not change its logical behavior, it only gives a more efficient form.
Final View
Boolean algebra is the language used to manipulate logical expressions in digital electronics. It is based on binary variables and a small set of operations, mainly AND, OR, and NOT. Its laws, especially complement rules, absorption rules, and De Morgan's theorems, allow complex logical expressions to be rewritten in simpler equivalent forms. This is essential for understanding and designing digital systems.