Boolean Algebra Calculator
Boolean algebra solver helps you to simplify Boolean expressions within simple steps of calculations. You can input logical expressions using variables and operators, and our Boolean expressions calculator displays a truth table, logical circuits, and a graph of the output.
What is Boolean Algebra?
“Boolean algebra is a branch of mathematics in which variables can take two values: false and true, which are indicated by 0 and 1, respectively. It involves basic logical operators such as AND, OR, and NOT that are used to process truth values.”
Boolean algebra is also known as binary algebra or logical algebra.
How to Use a Boolean Algebra Calculator?
Enter the Expression
Type your Boolean expression in the input box of the Boolean expressions simplifier
Use Special Symbols (if needed)
By using the left (◀) and right (▶) buttons, you can navigate and edit your expression easily.
- Use ! to insert a NOT gate
- Use + for OR operation
- Use direct letters like AB for AND (A AND B)
Click the CALCULATE Button
After typing your expression, click the "CALCULATE" button to simplify or evaluate the result.
What are Basic Logic Operators?
| Name |
Algebraic Function |
| AND |
F = A.B |
| OR |
F = A + B |
| NOT |
F = A |
| NAND |
F = (AB) |
| NOR |
F = (A + B) |
Example:
Find the truth table for the following Boolean expression:
(A + C) . B
Solution:
Step 1: Break the expression into smaller operations.
A + C = D
D . B = Output
Step 2: Solve these functions separately and combine them in one logic table.
| A |
B |
C |
A + C = D |
D . B |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
1 |
0 |
0 |
0 |
| 1 |
0 |
0 |
1 |
0 |
| 1 |
1 |
0 |
1 |
1 |
How to Simplify/Minify a Boolean Expression?
To simplify and minify the Boolean expression, we use a Boolean algebra simplifier calculator, which applies laws to reduce complex logic expressions. This helps define the operation of a digital logic circuit more efficiently, often with fewer variables and operations.
These Boolean algebra laws are given below:
Laws of Boolean Algebra:
1️⃣ Commutative Law:
The order in which two variables are combined using the AND (·) and OR (+) operations does not change the result.
- A + B = B + A
- A * B = B * A
2️⃣ Associative Law:
It is stated that the grouping of variables in an AND or OR operation does not affect the result of the expression.
- (A + B) + C = A + (B + C)
- (A * B) * C = A * (B * C)
3️⃣ Identity Law:
This law is stated as the combining of a variable with an identity element does not change its value.
4️⃣ Zero and One Law:
The zero and one law, also known as the Annulment, defines how the constants 0 and 1 interact with the AND and OR operations
5️⃣ Complement Law:
It states that a variable combined with its complement results in a fixed value. Variable AND equals 0 while OR equals 1.
6️⃣ Idempotent Law:
When the variable remains unchanged when it is combined with itself under AND or OR, resulting in the same variable.
7️⃣ Absorption Law:
The absorption laws in Boolean algebra simplify the algebraic expressions by absorbing one variable into another.
- A + (A * B) = A
- A * (A + B) = A
8️⃣ Distributive Law:
This law allows for the simplification of expressions involving the distribution of both AND and OR operations over other operations.
- A * (B + C) = (A * B) + (A * C)
- A + (B * C) = (A + B) * (A + C)
9️⃣ De Morgan's Law:
In Boolean algebra, De Morgan's Law states how negation distributes over AND and OR operations.
- (A + B)' = A' * B'
- (A * B)' = A' + B'
🔟 Double Negation Law:
The Double Negation Law states that if you negate a statement twice, you end up with the original statement.
What are the Applications of Boolean Algebra Calculator?
⁍ Digital electronics
⁍ Computer programming
⁍ Mathematics and set theory
⁍ Search engines and databases
⁍ Control systems
⁍ Networking (routing and switching)
⁍ Arithmetic logic units (ALUs)
⁍ Error detection and correction
⁍ Database query optimization
Truth Table for Binary Logical Operations
| p |
q |
F |
NOR |
↚ |
¬p |
↛ |
¬q |
XOR |
NAND |
AND |
XNOR |
q |
→ |
p |
← |
OR |
T |
| T |
T |
F |
F |
F |
F |
F |
F |
F |
F |
T |
T |
T |
T |
T |
T |
T |
T |
| T |
F |
F |
F |
F |
F |
T |
T |
T |
T |
F |
F |
F |
F |
T |
T |
T |
T |
| F |
T |
F |
F |
T |
T |
F |
F |
T |
T |
F |
F |
T |
T |
F |
F |
T |
T |
| F |
F |
F |
T |
F |
T |
F |
T |
F |
T |
F |
T |
F |
T |
F |
T |
F |
T |
| Com |
✓ |
✓ |
|
|
|
|
✓ |
✓ |
✓ |
✓ |
|
|
|
|
✓ |
✓ |
| Assoc |
✓ |
|
|
|
|
|
✓ |
|
✓ |
✓ |
✓ |
|
✓ |
|
✓ |
✓ |
| Adj |
F |
NOR |
↛ |
¬q |
↚ |
¬p |
XOR |
NAND |
AND |
XNOR |
p |
← |
q |
→ |
OR |
T |
| Neg |
T |
OR |
← |
p |
→ |
q |
XNOR |
AND |
NAND |
XOR |
¬q |
↛ |
¬p |
↚ |
NOR |
F |
| Dual |
T |
NAND |
→ |
¬p |
← |
¬q |
XNOR |
NOR |
OR |
XOR |
q |
↚ |
p |
↛ |
AND |
F |
| L id |
|
|
F |
|
|
|
F |
|
T |
T |
T,F |
T |
|
|
F |
|
| R id |
|
|
|
|
F |
|
F |
|
T |
T |
|
|
T,F |
T |
F |
|
FAQs:
What is the importance of this Boolean algebra calculator?
This Boolean algebra expression calculator is important because it helps to simplify logic expressions quickly, generate truth tables, reduce circuit complexity, save time, and effort.
What is a logic gate?
A logic gate is a fundamental building block of digital circuits that performs a Boolean function. It operates on one or more binary inputs (0s and 1s) and produces a single binary output.
Common types of logic gates include AND, OR, NOT, NAND, NOR, XOR, and XNOR.
What are minterms and maxterms?
These are both the fundamental concepts of Boolean algebra.
- Minterms: It is a product term (AND operation) where the function is true 1 in either its original or complemented form.
- Maxterms: It is a sum term (OR operation) where each variable also appears once, either complemented or not.
What are the symbols of Logic Operators?
Logical operators like:
- AND (∧ or &)
- OR (∨ or |)
- NOT (¬ or !)
- IMPLIES (→ or =>)
- XOR (⊕ or ^)
Reference:
Boolean algebra explained | source by Wikipedia
The Mathematics of Boolean Algebra (Stanford Encyclopedia of Philosophy) | Plato.Stanford.Edu.
Boolean laws – theorems | Goerge Boole (1854)-Tutorialspoint.Com
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