Boolean algebra is a division of mathematics that deals with operations on logical values and incorporates binary variables. Use our Boolean Algebra calculator for expression solving. Click below to utilise it right now.
What is Boolean Algebra?
Boolean algebra is a branch of algebra where the truth values, “ true “ and “ false “ are used as variable values. The truth value “ true “ is denoted by 1 while the truth value “ false “ is denoted by “ 0 “.
Truth Table
The truth table is a table that gives all the possible values of logical variables and the combination of the variables. It is possible to convert the Boolean equation into a truth table. In other words, a truth table is a mathematical table that lists the output of a particular digital logic circuit for all the possible combinations of its inputs. These truth tables can be used to deduce the logical expression for a given digital circuit, and are used extensively in Boolean algebra.
How is a truth table formed?
The following are the characteristics of a truth table that enable its formation –
- A truth table has one column for each input variable (commonly represented as A and B, x and y, or P and Q), and one final column showing all of the possible results of the logical operation that the table represents (for example, A AND B).
- Each row of the truth table contains one possible configuration of the input variables (for example, A = true [1 ] B = false [ 0 ] ), and the result of the operation for those values (continuing the example, A AND B=false [ 0 ] ).
Important Operations in Boolean Algebra
There are three important operations in Boolean algebra, namely –
- Conjunction
- Disjunction
- Negation
Conjunction
A conjunction is also known as AND operation. The symbol “ . “ is used to represent the conjunction between two variables A and B. This operation returns a true value only if both the input operands are true. The following will be the truth table for conjunction or AND operation between two variables A and B –
| A | B | A AND B |
| 1 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 0 |
Disjunction
Disjunction is also known as OR operation. The symbol “ + “ is used to represent the disjunction between two variables A and B. This operation returns a true value even if one of the input operands are true. The following will be the truth table for disjunction or OR operation between two variables A and B –
| A | B | A OR B |
| 1 | 1 | 1 |
| 1 | 0 | 1 |
| 0 | 1 | 1 |
| 0 | 0 | 0 |
Negation
Negation is also known as NOT operation. The symbol “ ‘ “ or “ – “ is used to represent the Negation or NOT of a variable A. This means that the negation of A is represented by A’. This operation returns a false value if the input value is true and a true value if the input value is false. The following will be the truth table for Negation or NOT operation for a variable A –
| A | A’ |
| 1 | 0 |
| 0 | 1 |
Laws of Boolean algebra
Below are some of the important laws of Boolean algebra –
Commutative law
According to this law –
- A + B = B + A
- A . B = B . A
Associative Law
According to this law –
- A + ( B + C ) = ( A + B ) + C
- A . ( B . C ) = ( A . B ) . C
Distributive Law
According to this law –
- A . ( B + C ) = ( A. B ) + ( A . C )
- A + ( B . C ) = ( A + B ) . ( A . C )
Identity Law
According to this law –
- A + 0 = A
- A . 1 = A
Idempotent Law
According to this law –
- A + A = A
- A . A = A
De Morgan’s Law
According to this law –
- ( A . B ) ‘ = A’ + B’
- ( A + B ) ‘ = A’ . B’
Boolean expressions and how to solve them?
A logical statement that results in a Boolean value, either be True or False, is a Boolean expression. Let us understand it through an example.
Suppose we have the Boolean expression A + B’
How will we solve it? Let us prepare a truth table for the same. We will have,
| A | B | B’ | A + B’ |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
Note that in the table above, we first calculated the value of B’ be performing the NOT operation. Thereafter we performed the OR operation between A and B’ to obtain the desired output for A + B’.
How to use the Boolean Algebra Calculator for solving Boolean expressions?
The following steps should be used to find the value of a Boolean expression using the Boolean algebra calculator –
Step 1 – The first step is to enter the expression in the “ Enter expression “ section of the Boolean algebra calculator. Let us take an example. Suppose we wish to solve the Boolean expression A + B + C. for doing so, we will enter this expression in the “ Enter expression “ section of the Boolean algebra calculator. Below is a snapshot of how the expression shall be entered –
Step 2 – Once we have entered the expression, the next step is to get the result. For doing so, we just need to click on the calculate button. As soon as we do so, we shall get the result on the right-hand side of the expression that we had entered in the previous step. Below is a snapshot of how the result will be displayed –
Now, let us compare this result with the one we would have got using Boolean laws. Let us make a truth table of A + B + C
| A | B | C | B + C | A + B + C |
| 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 |
| 1 | 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 1 |
| 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 0 | 1 | 1 |
| 0 | 0 | 1 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 |
If we compare both the results, we can see that we have obtained the same result through the calculator. In this way, we can use the Boolean algebra calculator to find the value of any Boolean expression.