Expression:
Solution:
Now we are solving above expression using boolean theorems:
The boolean algebra calculator is an expression simplifier for simplifying algebraic expressions. It is used for finding the truth table and the nature of the expression.
Follow the 2 steps guide to find the truth table using the boolean algebra solver.
Take help from sample expressions in the input box or have a look at the boolean functions in the content to understand the mathematical operations used in expressions.
Mathematics has different branches e.g algebra, geometry e.t.c. These branches are further divided into sub-branches. Boolean algebra is one such sub-branch of algebra.
It has two binary values including true and false that are represented by 0 and 1. Where 1 is considered as true and 0 is considered as false.
Boolean expressions are simplified to build easy logic circuits.
Boolean algebra has a set of laws or rules that make the Boolean expression easy for logic circuits. Through applying the laws, the function becomes easy to solve.
Here are the simplification rules:
Commutative law:
According to this law;
A + B = B + A
A.B = B.A
Associative law:
This law states;
A + ( B + C ) = ( A + B ) + C
A(B.C) = (A.B)C
Distributive law:
Using this law, we know;
A . ( B + C ) = ( A . B ) + ( A . C )
A + ( B . C ) = (A + B ) . (A + C )
Identity law:
By identity law:
A + 0 = A
A . 1 = A
Annulment law:
Here;
A . 0 = 0
A + 1 = 1
Idempotent law:
By this law:
A + A = A
A . A = A
There are some other rules but these six are the most basic ones.
Boolean algebra can be used on any of the systems where the machine works in two states. For example, the machines that have the option of “On” or “Off”.
Here are some of the real-time applications in our daily life that are using the concept of Boolean algebra:
Coffee/Tea machine
Rocket Launcher
Elevator for two floors
Car (Starting and turning off the engine)
Here is a table with Boolean functions and expressions:
| Name | Algebraic function |
|---|---|
| AND | F = A.B |
| OR | F = A+B |
| NOT | F = A |
| NAND | F = (A.B) |
| NOR | F = (A+B) |
| A | B | C | A+B | A+C | (A+B)(A+C) | BC | A+BC |
|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 |
| 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 |
| 1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Here is a truth table for all 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 | ||||||||||
Use the following rules and laws of boolean algebra to evaluate the boolean expressions:
| AND Form | OR Form | |
| Commutative Law | A.B=B. A | A + B = B + A |
| Associate Law | (A. B) . C = A. (B C) | (A + B) + C = A + (B + C) |
| Distributive Law | (A+B)+ C = (A+C). (B+C) | (A + B) C = (A. C) + (B C) |
| Identity Law | A. 1 A | A- + 0 = A |
| Zero and One Law | A. 0 = 0 | A+ 1 = 1 |
| Inverse Law | A.A' = 0 | A+ A' = 1 |
| Idempotent Law | A.A= A | A+A = A |
| Absorption Law | A(A+B) = A | A + A.B = A A+ A'B = A+B |
| DeMorgan's Law | (A. B)'=(A)'+ (B)' | (A+B)' = (A)' . (B)' |
| Double Complement Law | x = x |
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