Boolean Algebra – Its Definition, Gates, Theorems, and Examples

Boolean algebra is a mathematical technique used to solve problems in digital electronics, such as computer circuits. It is based on the algebra of sets, which is the mathematics of working with sets of objects.

Boolean algebra has its roots in the work of George Boole, a 19th-century mathematician who developed a system of algebra that could be used to solve problems in logic.

The term “Boolean algebra” was first coined by Claude Shannon, a 20th-century mathematician and electrical engineer who used Boolean algebra to simplify the design of digital circuits. Boolean algebra is used in many different fields, including computer science, engineering, and mathematics.

In this post, we will explain the term Boolean algebra along with examples.

## What Is Boolean Algebra?

Boolean algebra is the study of logical operations that take place within digital circuits. These operations can be used to simplify and optimize logic circuits, which can in turn improve the performance of these circuits.

Furthermore, the principles of Boolean algebra are also applied in other areas such as computer programming and database design. This means that Boolean algebra has many applications in the real world, including error detection and correction, data compression, and cryptography.

## Gates of Boolean Algebra

There are various types of gates that can be used in digital circuits. Each type of gate has its own set of properties and applications. Gates can be classified according to their functionality, such as OR, AND, NOR, etc.

### 1. OR Gate

OR gates are the simplest type of gate and are used to combine two bits of data into a single bit. The output of an OR gate is always either one or the other bit from the inputs.

### 2. AND Gate

AND gates are similar to OR gates in that they allow two bits of data to be combined together into a single bit. However, unlike OR gates, the output of an AND gate is always both bits from the inputs at once.

### 3. NOR Gate

NOR gates are used to create circuits that reverse the output of an AND gate. The output of a NOR gate is always zero if one input is true and one input is false.

## Theorems of Boolean Algebra

Boolean algebra is a mathematical discipline that deals with the analysis and synthesis of digital logic circuits. In Boolean algebra, there are three basic laws or rules which are known as axioms:

- The Commutative Law
- The Associative Law
- The Distributive Law

These three laws deal with the basic operations that can be performed on digital bits in a circuit. They allow for the construction of logical circuits and systems from these building blocks.

Additionally, these laws also provide a way to relate different bits of information within a system. For example, if we have two pieces of information, A and B, then we can say that A is distributively related to B if there exists a function “**f”** such that AB = f(A).

If you’re interested in learning more about Boolean Algebra or using it to solve problems in your field, try a Boolean algebra calculator.

## How to calculate Boolean algebra problems?

The problems of Boolean algebra can be solved either by using its laws or a truth table. Here are a few examples to understand the term Boolean algebra calculations.

**Example 1**

Calculate the Boolean algebra of the given expression either by using theorems or a truth table.

(C * A) + (A * B) (B + C)

**Solution **

**Step 1:** Write the given Boolean expression.

(C * A) + (A * B) (B + C)

Now calculate the above expression according to the theorems of Boolean algebra.

**Step 2:** Apply the sum of product law.

(C * A) + (A * B) (B + C) = AC + AB (B + C)

(C * A) + (A * B) (B + C) = AC + ABB + ACB

**Step 3:** Now apply the Idempotent Law (AA = A).

(C * A) + (A * B) (B + C) = AC + AB + ACB

**Step 4:** Factorize the above expression.

= AC + AB + ACB

= AC (B + 1) + AB

**Step 5:** Now apply the identity law (A + 1 = 1) and (A.1 = 1).

= AC (1) + AB

= AC + AB

It can also be written as:

= A * (B + C)

Hence the given Boolean expression is equal to:

(C * A) + (A * B) (B + C) = A * (B + C)

**Alternatively**

Now we will solve the given expression according to the truth table.

**Step 1:** First of all, identify the rows of the table that would be involved by taking the total number of digits in the expression without repetition.

Terms in the given expression = n = 3

According to formula

2^{n} = 2^{3} = 2 x 2 x 2 = 8

Hence, there will be 8 rows in the truth table.

**Step 2:** Now make a truth table according to the given Boolean expression.

A |
B |
C |
A * B |
C * A |
B + C |
(A * B) (B + C) |
(C * A) + (A * B) (B + C) |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 |

0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 |

0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 |

1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |

1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

**s**

A Boolean algebra calculator is a helpful resource to calculate the problems of binary algebra according to the laws or truth table to avoid time-consuming calculations.

**Example 2**

Calculate the Boolean algebra of the given expression either by using theorems or a truth table.

(A + B)C + B(A + C)

**Solution **

**Step 1:** Write the given Boolean expression.

(A + B)C + B(A + C)

Now calculate the above expression according to the theorems of Boolean algebra.

**Step 2:** Apply the sum of product law.

(A + B)C + B(A + C) = (A + B)C + BA + BC

(A + B)C + B(A + C) = CA + CB + BA + BC

**Step 3:** Now factorize the above expression.

(A + B)C + B(A + C) = CA + B(C + A + C)

**Step 4:** Now apply the Idempotent Law (A + A = A)

(A + B)C + B(A + C) = CA + B(C + A)

It can also be written as:

(A + B)C + B(A + C) = CA + BC + AB

Hence the given Boolean expression is equal to CA + BC + AB

**Alternatively**

Now we will solve the given expression according to the truth table.

**Step 1:** First of all, identify the rows of the table that would be involved by taking the total number of digits in the expression without repetition.

Terms in the given expression = n = 3

According to formula

2^{n} = 2^{3} = 2 x 2 x 2 = 8

Hence, there will be 8 rows in the truth table.

**Step 2:** Now make a truth table according to the given Boolean expression.

A |
B |
C |
A + B |
A + C |
(A + B) C |
B (A + C) |
(A + B)C + B(A + C) |

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 | 0 | 0 | 0 |

1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 |

1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

## Conclusion

Boolean algebra is a method that can be used to solve problems in many different areas, including computer science, engineering, and mathematics. It is based on the algebra of sets, which is the mathematics of working with sets of objects. You can grab the basics of this topic from this post.