# Boolean‌ ‌algebra‌

The algebraic equations that contain the actual values of their variables are called Boolean algebra. The truth of an equation is denoted by 1 or 0. It is also known as logical algebra or binary algebra. It is used to analyze digital gates and digital circuits by system engineers. The critical operations in Boolean algebra include – negation (-), conjunction (^), and disjunction (v).

## Boolean algebra operations

The basic operations in a Boolean algebra can be denoted as-

 Operator Symbol Precedence NOT ‘ (or) ¬ Highest AND . (or) ∧ Middle OR + (or) ∨ Lowest

For example, suppose X and Y are two Boolean variables, then one can define the three operations as-

• X conjunction Y or X AND Y, satisfies X ∧ Y = True, if X = Y = True or else X ∧ Y = False.
• X disjunction Y or X OR Y, satisfies X ∨ Y = False, if X = Y = False, else X ∨ Y = True.
• Negation X or ¬X satisfies ¬X = False, if X = True and ¬X = True if X = False

## Boolean expression

A logical expression that is either true or false is known as a Boolean expression. Sometimes in place of true or false, Yes, No, etc., can also be used. They are the expressions with logical operators, like AND, OR, XOR, and NOT. Thus, if we say G XOR H, then this is a Boolean expression.

## Truth table of Boolean algebra

 A B A ∧ B A ∨ B True True True True True False False True False True False True False False False False

## Rules of Boolean algebra

1). Each variable in a Boolean expression can have only two values, i.e., 1 for high and 0 for low.

2). The complement of a variable is denoted by the bar over the variable.

3). OR expressions are represented by + sign between them. Whereas AND operations are denoted by x or . between them. For example, K OR L can be represented as K+L, and K AND L is represented as K.L or K x L.

## Laws of Boolean algebra

1). Commutative law – Boolean is commutative over one another, conditioned, they do not change the value of the operated gate. For example, K.L = L.K

2). Distributive law – Boolean expression is distributive over one another. For example, X (Y+Z) = XY + XZ.

3). Associative law – Boolean expression is associative. For example, E. (F.G) = (E.F) .G

4). AND law – The following expressions denote AND law-

a. A.0 = 0

b. A.1 = A

c. A.A = A

d. A.A = 0

5). OR law – The following expressions denoted OR law-

a. A + A = 1

b. A+A = A

c. A+1= 1

d. A+0 = A

6). Inversion law – This uses NOT operation. This is denoted as A + A = 1

## Boolean algebra theorems

1). De Morgan’s first law – The truth table of De Morgan’s first law is given by-

 A B A' B' (A.B)' A'+B' 0 0 1 1 1 1 0 1 1 0 1 1 1 0 0 1 1 1 1 1 0 0 0 0

2). De Morgan’s first law – The truth table of De Morgan’s first law is given by-

 A B A' B' (A.B)' A'+B' 0 0 1 1 1 1 0 1 1 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0
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