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# Properties of Boolean Algebra

• Difficulty Level : Easy
• Last Updated : 28 Dec, 2022

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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