Boolean algebra is the branch of algebra in which the values of the variables are truth-values, true and false, usually represented as 1 and 0. Boolean algebra is used for analyzing and simplifying digital circuits or digital gates. Boolean algebra is also known as binary or logical algebra because it involves logical operations and binary variables. It plays a vital role in digital electronics, modern programming languages, and set theory.

Boolean algebra is far different from the elementary algebra. Boolean algebra deals with logical operations such as conjunction (∧), disjunction (∨), and negation (¬). However, elementary algebra uses addition, subtraction, division, and multiplication operations.

George Boole (1815 – 1864) introduced the idea of Boolean algebra in his book “The Mathematics Analysis of Logic”. Further, he expanded this concept in his book “An Investigation of Laws of Thought”.

**Boolean Operators**

We will discuss the three most common Boolean operators, which are given below:

- Negation or NOT operator
- Conjunction or AND operator
- Disjunction or OR operator

p | ¬p |

T | F |

F | T |

**Negation or NOT operator**

p | q | p ∧ q |

T | T | T |

T | F | F |

F | T | F |

F | F | F |

A proposition is a declarative statement that may be true or false but cannot be both simultaneously. If p is any proposition, its negation is denoted by ¬p, read “not p”. It is an operator that reverses the nature of value, i.e. if p is true, ¬p is false, and if p is false, ¬p is true. Table 1 shows the truth table of negation.

**Conjunction (or AND operator)**

The conjunction of two propositions p and q is denoted by ‘p ∧ q’ or ‘p. q’. The conjunction ‘p ∧ q’ is only true when the components of p and q are true. The truth table of conjunction (p ∧ q) is shown in Table 2.

**Disjunction or OR operator**

p | q | p ∨ q |

T | T | T |

T | F | T |

F | T | T |

F | F | F |

It is written as p ∨ q (or p + q). When at least one component of p or q is true, the disjunction (p ∨ q) is true. When components of p and q are false, it is false. The truth table of disjunction (p ∨ q) is shown in Table 3.

**Rules for Boolean Algebra**

Here are some important rules for Boolean Algebra:

- Variables in Boolean algebra have only two values: 1 for True, and 0 for False.
- The negation or complement of a variable is indicated by an overbar (-) or dash. For example, B’ represents the complement of variable B.
- The OR operation between variables is denoted by a plus sign. For example, A + B + C signifies the OR operation between variables A, B, and C.
- The logical AND operation between variables is represented by a dot (.) or simply writing the variables together. For example, A . B. C or ABC denotes the logical AND operation between variables A, B, and C.

**Laws of Boolean Algebra**

The laws of Boolean algebra allow us to simplify complex Boolean expressions. Important laws of Boolean Algebra are discussed in the following.

**Commutative Law**

Commutative law states that the order of two separate propositions does not affect the result of the logical operation. This law applies to both OR and AND operators.

- A + B = B + A
- A . B = B . A

**Associative Law**

This law is for several variables. According to this law, there is no change in the results if a grouping of expressions is changed.

- (A + B) + C = A + (B + C)
- (A . B) . C = A . (B . C)

**Distributive Law**

This law is discussed in two ways, i.e., “AND over OR” and “OR over AND”.

- A . (B + C) = (A . B) + (A . C) (AND over OR)
- A + (B . C) = (A + B) . (A + C) (OR over AND)

**Identity Law**

If a variable is OR’ed with a False, the result is always equal to the variable. If a variable is AND’ed with true, the result is always equal to that variable.

- A + 0 = A
- A . 1 = A

**Double Negation Law:**

- (A’)’ = A

**Complement Laws:**

- A + A’ = 1 (True)
- A . A’ = 0 (False)
- 1 + A’ = 1

**Idempotent Laws:**

- A + A = A
- A . A = A

**Null Elements Laws:**

- A + 1 = 1
- A . 0 = 0

**De Morgan’s Theorems**

De Morgan’s theorems provide a way to simplify an expression involving negation (NOT) operators. They state that the complement of logical conjunction (AND) is equivalent to the logical disjunction (OR) of the complement of individual statements, and vice versa. These thermoses are often used in reducing the complexity of Boolean expressions.

- (A + B)’ = A’. B’
- (A . B)’ = A’ + B’

**Solved Examples of Boolean algebra**

**Example 1:**

Write the following expression in simplest form by using the laws of Boolean algebra.

AB + B

**Solution: **

By using the distributive laws, we can write the given expression as

AB + B = B (A + 1)

∴ A + 1 = 1 (for any value of A)

= B (1) = B

Therefore, the simplest form of (AB + B) is B.

**Example 2:**

Solve the following expression.

B + (A. B)’

**Solution: **

∴ (A . B)’ = A’ + B’ (De Morgan’s Law)

B + (A. B)’ = B + ( A’ + B’)

= B + (B’ + A’) (By using Commutative law)

= (B + B’) + A’ (By using Associative law)

∴ (B + B’) = 1 (Complement law)

= 1 + A’ = 1

Therefore, B + (A. B)’ = 1

A Boolean algebra calculator by Allmath (https://www.allmath.com/boolean-algebra-calculator.php) can be used to solve Boolean problems according to its laws with steps.

**Example 3:**

Draw the truth table for (A. B)’

**Solution: **

A | B | A.B | (A.B)’ |

1 | 1 | 1 | 0 |

1 | 0 | 0 | 1 |

0 | 1 | 0 | 1 |

0 | 0 | 0 | 1 |

**Conclusion:**

In this article, we debated Boolean algebra, which deals with logical operations or binary variables. We have discussed the three common Boolean operators. We talked about the rules and laws of Boolean algebra. We examined De Morgan’s theorem. We did solve many examples of Boolean algebra. After understanding this article, you can simplify any expression with the help of Boolean algebra laws and draw truth tables easily.

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