Properties of Boolean Algebra

Last Updated : 23 Jul, 2023

Switching algebra is also known as Boolean Algebra. It is used to analyze digital gates and circuits It is logical to perform a mathematical operation on binary numbers i.e., on ‘0’ and ‘1’. Boolean Algebra contains basic operators like AND, OR, and NOT, etc. Operations are represented by ‘.’ for AND , ‘+’ for OR. Operations can be performed on variables that are represented using capital letters eg ‘A’, ‘B’ etc.

Properties of switching algebra:

Annulment law – a variable ANDed with 0 gives 0, while a variable ORed with 1 gives 1, i.e.,

```A.0 = 0
A + 1 = 1 ```

Identity law – in this law variable remain unchanged it is ORed with ‘0’ or ANDed with ‘1’, i.e.,

```A.1 = A
A + 0 = A ```

Idempotent law – a variable remains unchanged when it is ORed or ANDed with itself, i.e.,

```A + A = A
A.A = A ```

Complement law – in this Law if a complement is added to a variable it gives one, if a variable is multiplied with its complement it results in ‘0’, i.e.,

```A + A' = 1
A.A' = 0 ```

Double negation law – a variable with two negations, its symbol gets cancelled out and original variable is obtained, i.e.,

`((A)')'=A `

Commutative law – a variable order does not matter in this law, i.e.,

```A + B = B + A
A.B = B.A  ```

Associative law – the order of operation does not matter if the priority of variables are the same like ‘*’ and ‘/’, i.e.,

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

Distributive law – this law governs the opening up of brackets, i.e.,

```A.(B+C) = (A.B)+(A.C)
(A+B)(A+C) = A + BC ```

Absorption law –:-This law involved absorbing similar variables, i.e.,

```A.(A+B) = A
A + AB = A
A+ A'B = A+B
A(A' + B) = AB```

De Morgan law the operation of an AND or OR logic circuit is unchanged if all inputs are inverted, the operator is changed from AND to OR, and the output is inverted, i.e.,

```(A.B)' = A' + B'
(A+B)' = A'.B' ```

Consensus theorem:

`AB + A'C + BC = AB + A'C`

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