Boolean algebra is a well-known branch of algebra that deals with the truth values i.e., true or false. The digital circuits and the digital gates are simplified with the help of this branch of algebra. The modern programming languages and the modern development of digital electronics.
Boolean algebra is a well-known technique that is widely used in mathematics, computer, set theory, and statistics for various purposes.
What is Boolean Algebra?
In Boolean algebra variables are studied only in binary form. A Boolean variable is typically defined as either true or false. 1 is represented as true and 0 is represented as false. There are also more complex interpretations of variables, such as those found in set theory. Binary algebra is another name for Boolean algebra.
The most common operations that are performed in binary algebra are negation, conjunction, and disjunction. This kind of algebra is not similar to elementary algebra. The rules of addition and subtraction are different in this type of algebra.
Operations of Boolean Algebra
Three main operations are involved in this type of algebra.
- Conjunction (^)
- Disjunction (v)
- Negation (-)
These Boolean operators are frequently used to evaluate the sum, difference, and transpose of two variables.
1. Conjunction (^)
A conjunction is an operation of binary xalgebra used to multiply the variables. This operation is said to be the AND operation and is denoted by the “^” symbol. The values are true only when both values are true otherwise the values are false. the values can be written as “A.B” or A ^ B.
For example
| P | Q | P^Q |
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
2. Disjunction (v)
Disjunction is an operation of binary algebra used to add variables. This operation is said to be the OR operation and is denoted by the “v” symbol. The values are false only when both the values are false otherwise the values are true. The values can be written as “A + B” or A v B.
| P | Q | PvQ |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
3. Negation (-)
Negation is an operation of Boolean algebra that reverts the truth values. If true, it transposes to false and if false, it converts to true. The values of negation can be represented as –A, ¬A, or bar A.
| P | -P |
| 0 | 1 |
| 1 | 0 |
How to calculate the expression of Boolean Algebra?
Here are a few solved examples to learn how to evaluate the problems of Boolean algebra.
By using Laws of Boolean Algebra
Example 1
Find the expression of Boolean algebra with the help of the laws of Boolean algebra.
(-P + Q) + PQ + (Q + (-R))
Solution
Step 1: First of all, write the expression of binary algebra in the simplest form.
(-P + Q) + PQ + (Q + (-R))
-P + Q + PQ + Q + (-R)
Step 2: Apply the idempotent law of binary algebra with respect to addition (P + P) = P
-P + PQ + Q + Q + (-R)
-P + PQ + Q + (-R)
Step 3: Now apply the identity law of Boolean algebra with respect to addition (P + 1 = 1).
-P + (P + 1) Q + (-R)
-P + (1) Q + (-R)
-P + Q + (-R)
Step 4: Now write the result with an expression.
(-P + Q) + PQ + (Q + (-R)) = -P + Q + (-R)
Example 2
Find the expression of Boolean algebra with the help of the laws of Boolean algebra.
(P * Q) + P * R + Q[-P + (Q + R)]
Solution
Step 1: First of all, write the expression of binary algebra in the simplest form.
(P * Q) + P * R + Q[-P + (Q + R)]
(PQ) + PR + Q[-P + P + (Q + R)]
Step 2: Make the factors of the given expression.
(Q + R)P + Q(-P + P + Q + R)
Step 3: Now apply the law of complement to the above expression (P+(-P) = 1)
(Q + R)P + Q(1 + Q + R)
Step 4: Now apply the identity law of Boolean algebra with respect to addition (Q + 1 = 1).
(Q + R)P + Q(1 + R)
Step 5: Now apply the identity law of Boolean algebra with respect to addition (R + 1 = 1).
(Q + R)P + Q(1)
(Q + R)P + Q
Step 6: Now write the result with an expression.
(P * Q) + P * R + Q[-P + (Q + R)] = (Q + R)P + Q
Try a boolean algebra calculator by AllMath [https://www.allmath.com/boolean-algebra-calculator.php] to solve the boolean algebra problems without any difficulty according to the laws.
By using the truth table
Now we’ll solve the above examples through truth tables.
Example 1
Find the expression of Boolean algebra with the help of the truth table.
(-P + Q) + PQ + (Q + (-R))
Solution
Step 1: Make 23 = 8 rows and write the truth table according to the given expression and get the result.
| P | Q | R | -P | -R | PQ | -P + Q | Q + (-R) | (-P + Q) + PQ | (-P + Q) + PQ + (Q + (-R)) |
| 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 |
| 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 |
| 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 |
| 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |
Example 2
Find the expression of Boolean algebra with the help of the truth table.
(P * Q) + P * R + Q[-P + (Q + R)]
Solution
Step 1: Make 23 = 8 rows and write the truth table according to the given expression and get the result.
| P | Q | R | -P | PR | PQ | PQ + PR | Q + R | -P + (Q + R) | Q[-P + (Q + R)] | PQ + PR + Q[-P + (Q + R)] |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 |
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Wrap Up
Boolean algebra is a well-known technique of algebra used to deal with the binary variables zero and one for false and true values respectively. The laws of Boolean algebra and the truth table play a vital role in the calculations of Boolean algebra.