Introduction to Boolean algebra: Definition, gates, and examples
Introduction to Boolean algebra: Definition, gates, and
examples
In algebra, Boolean algebra is frequently used to perform logical gates like AND, OR, and NOT. It is generally a branch of algebra that is used to simplify the circuits and gates by using 0 & 1 integers. It is also used in computers, mathematical logic, graph theory, and set theory.
The logical operations of this branch of algebra can be identified by using base 2 integers (0, 1). In this article, we’ll learn the definition, rules(laws), and examples of Boolean algebra.
Boolean algebra
In mathematics, Boolean algebra is a branch of algebra that used binary digits to represent the gates of mathematical logic. The binary digit zero is used for false and one is used for true in this branch of algebra.
It deals with three main operators such as:
- Conjunction
- Disjunction
- Negation
The above operators of Boolean algebra used normal mathematical symbols such as +, -, x, or /. But the working of these math symbols is not the same in Boolean algebra. It used a truth table to show all the logic operations and combinations.
The working of Boolean operators plays a vital role in solving its problems and making the solutions. Let’s discuss the working of these operators briefly.
Conjunction (AND)
The conjunction used 0 & 1 digits and can be expressed as 1 only when both the terms are 1 otherwise it gives zero. In simple words, the AND gate is the product of the given terms. It is denoted by the ^ symbol.
0 * 0 = 0
0 * 1 = 0
1 * 0 = 0
1 * 1 = 1
| S | T | S.T |
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Disjunction (OR)
The disjunction used 0 & 1 digits and can be expressed as 0 only when both the terms are 0 otherwise it gives one. In simple words, the OR gate is the sum of the given terms. It is denoted by the v symbol.
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 1
| S | T | S + T |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
Negation (NOT)
The negation used 0 & 1 digits and can be expressed as 1 when the given term is 0 and gives 0 when the given term is 1. In simple words, the NOT gate transpose the given terms. It is denoted by a bar on the digit.
1 = 0
0 = 1
| S | S– |
| 0 | 1 |
| 1 | 0 |
NAND and XOR gates
There are two other logic gates used in the Boolean algebra known as the NAND gate and XOR gate. The NAND gate is the combination of AND gate and NOT gate. In other words, the NAND gate is opposite to AND gate.
0 * 0 = 1
0 * 1 = 1
1 * 0 = 1
1 * 1 = 0
| S | T | (S.T)– |
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
While XOR gate is the combination of NOT gate and OR gate.
0 + 0 = 1
0 + 1 = 0
1 + 0 = 0
1 + 1 = 0
| S | T | (S + T)— |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |
The rows in the truth tables of these gates can be decided by the terms of the given variables after applying a formula 2n. where n is the number of given terms. You can use a Boolean algebra simplifier to get the result according to the operators of Boolean.
How to calculate the problems of Boolean algebra?
By using the formulas and gates, the problems of Boolean algebra can be solved easily. Let us take a few examples of Boolean algebra.
Example 1
Find the truth table of the given expression by using the gates of Boolean algebra.
(S + T) + (T + Q) * S
Solution
Step 1: Write the algebraic expression.
(S + T) + ((T + Q) * S)
Step 2: By taking the total number of terms identify the rows of the truth table.
Term in the given expression = n = 3
According to formula
2n = 23 = 2 x 2 x 2 = 8
Hence, there will be 8 number of rows in the truth table.
Step 3: According to the given terms make the truth table for three variables.
| S | T | Q | S + T | T + Q | (T + Q) * S | (S + T) + (T + Q) * S |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 | 1 |
| 0 | 1 | 0 | 1 | 1 | 0 | 1 |
| 0 | 1 | 1 | 1 | 0 | 0 | 1 |
| 1 | 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 |
You can also use a Boolean algebra calculator to verify the result. Follow the below steps to use this tool.
Step 1: Enter the expression
Step 2: Click calculate
Step 3: The result will be shown below the calculate button in a fraction of seconds.
Example 2
Find the truth table of the given expression by using the gates of Boolean algebra.
(F * G) + ((G * H) + G)
Solution
Step 1: Write the algebraic expression.
(F * G) + ((G * H) + G)
Step 2: By taking the total number of terms identify the rows of the truth table.
Term in the given expression = n = 3
According to formula
2n = 23 = 2 x 2 x 2 = 8
Hence, there will be 8 number of rows in the truth table.
Step 3: Make a truth table.
| F | G | H | F * G | G * H | (G * H) + G | (F * G) + ((G * H) + G) |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 0 | 0 | 1 |
| 1 | 1 | 0 | 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Summary
In this post, we have learned about the definition, logic gates, and examples of Boolean algebra. Now you are witnessed that making truth tables is not a difficult topic. You can grab all the basics of Boolean just by learning the above operators and examples.
In algebra, Boolean algebra is frequently used to perform logical gates like AND, OR, and NOT. It is generally a branch of algebra that is used to simplify the circuits and gates by using 0 & 1 integers. It is also used in computers, mathematical logic, graph theory, and set theory.
The logical operations of this branch of algebra can be identified by using base 2 integers (0, 1). In this article, we’ll learn the definition, rules(laws), and examples of Boolean algebra.
Boolean algebra
In mathematics, Boolean algebra is a branch of algebra that used binary digits to represent the gates of mathematical logic. The binary digit zero is used for false and one is used for true in this branch of algebra.
It deals with three main operators such as:
- Conjunction
- Disjunction
- Negation
The above operators of Boolean algebra used normal mathematical symbols such as +, -, x, or /. But the working of these math symbols is not the same in Boolean algebra. It used a truth table to show all the logic operations and combinations.
The working of Boolean operators plays a vital role in solving its problems and making the solutions. Let’s discuss the working of these operators briefly.
Conjunction (AND)
The conjunction used 0 & 1 digits and can be expressed as 1 only when both the terms are 1 otherwise it gives zero. In simple words, the AND gate is the product of the given terms. It is denoted by the ^ symbol.
0 * 0 = 0
0 * 1 = 0
1 * 0 = 0
1 * 1 = 1
| S | T | S.T |
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Disjunction (OR)
The disjunction used 0 & 1 digits and can be expressed as 0 only when both the terms are 0 otherwise it gives one. In simple words, the OR gate is the sum of the given terms. It is denoted by the v symbol.
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 1
| S | T | S + T |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
Negation (NOT)
The negation used 0 & 1 digits and can be expressed as 1 when the given term is 0 and gives 0 when the given term is 1. In simple words, the NOT gate transpose the given terms. It is denoted by a bar on the digit.
1 = 0
0 = 1
| S | S– |
| 0 | 1 |
| 1 | 0 |
NAND and XOR gates
There are two other logic gates used in the Boolean algebra known as the NAND gate and XOR gate. The NAND gate is the combination of AND gate and NOT gate. In other words, the NAND gate is opposite to AND gate.
0 * 0 = 1
0 * 1 = 1
1 * 0 = 1
1 * 1 = 0
| S | T | (S.T)– |
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
While XOR gate is the combination of NOT gate and OR gate.
0 + 0 = 1
0 + 1 = 0
1 + 0 = 0
1 + 1 = 0
| S | T | (S + T)— |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |
The rows in the truth tables of these gates can be decided by the terms of the given variables after applying a formula 2n. where n is the number of given terms. You can use a Boolean algebra simplifier to get the result according to the operators of Boolean.
How to calculate the problems of Boolean algebra?
By using the formulas and gates, the problems of Boolean algebra can be solved easily. Let us take a few examples of Boolean algebra.
Example 1
Find the truth table of the given expression by using the gates of Boolean algebra.
(S + T) + (T + Q) * S
Solution
Step 1: Write the algebraic expression.
(S + T) + ((T + Q) * S)
Step 2: By taking the total number of terms identify the rows of the truth table.
Term in the given expression = n = 3
According to formula
2n = 23 = 2 x 2 x 2 = 8
Hence, there will be 8 number of rows in the truth table.
Step 3: According to the given terms make the truth table for three variables.
| S | T | Q | S + T | T + Q | (T + Q) * S | (S + T) + (T + Q) * S |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 | 1 |
| 0 | 1 | 0 | 1 | 1 | 0 | 1 |
| 0 | 1 | 1 | 1 | 0 | 0 | 1 |
| 1 | 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 |
You can also use a Boolean algebra calculator to verify the result. Follow the below steps to use this tool.
Step 1: Enter the expression
Step 2: Click calculate
Step 3: The result will be shown below the calculate button in a fraction of seconds.
Example 2
Find the truth table of the given expression by using the gates of Boolean algebra.
(F * G) + ((G * H) + G)
Solution
Step 1: Write the algebraic expression.
(F * G) + ((G * H) + G)
Step 2: By taking the total number of terms identify the rows of the truth table.
Term in the given expression = n = 3
According to formula
2n = 23 = 2 x 2 x 2 = 8
Hence, there will be 8 number of rows in the truth table.
Step 3: Make a truth table.
| F | G | H | F * G | G * H | (G * H) + G | (F * G) + ((G * H) + G) |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 0 | 0 | 1 |
| 1 | 1 | 0 | 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Summary
In this post, we have learned about the definition, logic gates, and examples of Boolean algebra. Now you are witnessed that making truth tables is not a difficult topic. You can grab all the basics of Boolean just by learning the above operators and examples.
