Nowadays, we are surrounded by electronic equipment that is controlled by Boolean operations. Life appears to be impossible without these tools.
The sole mathematical concept that is crucial for someone who wants to study electronics and fields connected to logical electronic devices is a Boolean algebra.
A collection of expressions with only two possible states—true or false—were created by the mathematician George Boole in the middle of the nineteenth century.
In this post, we will cover the structure and working of circuits which execute the properties of basic Boolean expressions.
What is Boolean Algebra?
“Boolean Algebra is a creative and helpful method for simplifying digital circuits used in microprocessors.”
Or
“In mathematics, the algebra that can maintain an electronic circuit in one of two states, either 0 or 1, is called Boolean algebra.”
Electronic circuits utilize a variety of words, including literal and variable complements. A variable is a symbol that can have one of two values, such as 0 or 1. Its complement, which can be read as not A, is the inverse of this variable.
The variable that results after the compliment is known as literal.
What are Boolean addition, multiplication and Boolean inversion?
Logic gates are the building blocks of digital circuitry. Logic gates are basic digital components that perform basic logical operations. All logic gates have at least one input and one output, and they can only be in one of two states.
Here 0 characterizes the non-electrical state and 1 characterizes an electrical state. In terms of how they relate to logic gates like AND, OR, etc., Boolean expressions were first described.
Boolean Addition
In digital logic circuits, an OR gate performs the sum operation, but in Boolean algebra, the term addition refers to the sum of variables. Evidently, the only two conceivable states for sum terms are 0 and 1.
If one or more of the terms’ variables equals 1, the addition term will also equal 1. If all of the variables are 0, the addition term will also be 0.
Some examples of sum term are G+H, H+J or G+H+J.
Boolean Multiplication
The multiplication operation is carried out by an AND gate in digital logic circuits, however, in Boolean algebra, the term “multiplication” refers to the product of variables. It appears that there are just two possible states for product terms: 0 and 1.
The multiplication term will also equal 0 if one or more of the term’s variables are set to 0. The product term will also be 1 if all of the variables are 1.
GH, HJ, or GHJ are a few examples of product terms.
Boolean Inversion
In digital logic circuits, a NOT gate performs the inversion operation; however, in Boolean algebra, the term “inversion” refers to the complement of a variable. In this instance, a single input and single output result in one of two stable states.
If the input variable is set to one, the inversion will be zero, and if the variable is set to zero, the output will also be one.
Why the truth tables are created for Boolean expressions?
Truth tables are tables that show the relationship between the input and output of binary variables for each logical gate. Truth tables provide a concise explanation of how two logical conditions may be combined using AND, OR, and NOT or any other binary operation combination.
If we consider an OR gate, then the true condition at the output can only be carried out if one of its inputs is true, according to logic.
| G | H | X= G+H | Statement |
| 0 | 0 | 0 | F |
| 1 | 0 | 1 | T |
| 0 | 1 | 1 | T |
| 1 | 1 | 1 | T |
We learn how to negate a Boolean expression and what that entails from DE Morgan’s Laws. We frequently enter values into truth tables as either 1 or 0. 1 is true and 0 is false, and they may both be used to represent True and False. Several rows and columns make up a table. The logical variables and combinations are shown in the top row, descending in complexity from the final function.
How to make a truth table using Boolean Algebra?
Example 1
Create the truth table IF the Boolean expression connecting the two variables G and H is (-G*-H).
Solution: Manual method
Step 1: First, determine G’s negative complement, or -G.
| G | -G |
| 0 | 1 |
| 0 | 1 |
| 1 | 0 |
| 1 | 0 |
Step 2: Then calculate the complement of H.
| H | -H |
| 0 | 1 |
| 1 | 0 |
| 0 | 1 |
| 1 | 0 |
Step 3: Now make a truth table for expression (-G*-H).
| -G | -H | (-G*-H) |
| 1 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 0 |
A Boolean algebra calculator can resolve the difficulty of making truth table with just a single click. Utilizing a scientific approach enables us to solve issues quickly and successfully.
Example 2
Create the truth table IF the Boolean expression connecting the two variables L and M is (L*-M) + (-L*M).
Solution: Manual method
Step 1: First, determine L’s negative complement, or -L.
| L | -L |
| 0 | 1 |
| 0 | 1 |
| 1 | 0 |
| 1 | 0 |
Step 2: Calculate the determinant of M.
| M | -M |
| 0 | 1 |
| 1 | 0 |
| 0 | 1 |
| 1 | 0 |
Step 3: Now calculate the expressions (L*-M) and (-L*M).
| L | -L | M | -M | (L*-M) | (-L*M) |
| 0 | 1 | 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 | 1 |
| 1 | 0 | 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 | 0 | 0 |
Step 4: Now complete the table by connecting (L*-M) and (-L*M) through an OR gate.
| L | -L | M | -M | (L*-M) | (-L*M) | (L*-M) + (-L*M) |
| 0 | 1 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 1 | 1 | 0 | 1 |
| 1 | 0 | 1 | 0 | 0 | 0 | 0 |
Hence the truth table for a given Boolean expression has been created.
Wrap Up
We have discussed the principles of Boolean algebra, including its applications, significance, and operations, clearer. Moreover, you may generate a truth table for any Boolean expression using one of the two methods discussed above.