As this lesson you’ll able to learn **what is Boolean algebra**,** how we Simplify Boolean algebra**, **Basic and Axiomatic definition of Boolean algebra** and example. Let’s continue..

**Boolean Algebra** help to reduce the complex logical expression to simpler expression which saves the price of the unnecessary gates, reduce the number of gates, reduce the power consume and the amount of space the logic gates.

**George Bool in 1854** build laws of Boolean Algebra to reduce Complex logical expression in simpler form in digital logic or mathematics is known as** Boolean Algebra.**

**Basic definition Boolean algebra**–

**Boolean algebra** like other deductive mathematical system may be defined any with the set of elements, a set of operators and no. of unproved axioms and postulates. A set of element is any collection of object having a common property. A binary operator is a rule that assigns to each pair of elements to get a unique element.** The most common postulates to Formulate various Algebric structure are:**

##### Closure:

A set s is closed with respect to o binary operator of for every pair of elements of s, the binary operator specifies a rule for obtaining unique element of s**for example**

S={1,2,3,4,5}

a=2,b=3

a+b=2+3 = 5 = c (say)

$\mathrm{a},\mathrm{b},\mathrm{c}\in \mathrm{s}$

a-b=2-3=-1=c (say)

$\mathrm{a},\mathrm{b},\mathrm{c}\notin \mathrm{s}$

for the above example it is clear that gets is closed with respect to binary operator “+” and set s is not closed with respect to binary operator “-”

##### Associative law

A binary operator’*’ on a jet s´is said to be associative whenever (x*y)*z=x*(y*z) where, $\mathrm{x},\mathrm{y},\mathrm{z}\notin \mathrm{s}$##### Commutative law :

A binary operator ‘*’ a set s is said to be be commutative whenever x*y=y*x where, $\mathrm{x},\mathrm{y}\in \mathrm{s}$##### Identity element:

A set s is said to have identity element with respect to binary operator’*’ on set S, If there exist an element $\mathrm{e}\in \mathrm{s}$ with the property.e*x=x*e=x

##### Inverse element:

A set s having identity element e with respect to binary operator’*’ is said to have on inverse whenever for every x ∈ s there exist an element y ∈ s sucks that x*y=e=y*x#### Axiomatic definition of Boolean algebra:

Boolean Algebra is a algebric structure defined on any set of elements an together with two binary operators ‘+’ and ‘.’ provided the following postulates that are satisfied.

**Laws of Boolean Algebra for simplify**

- Closure with respect to operator
**+**and Closure with respect to operator**(.)**. - An identity element with respect to + designated by 0: X+0=0+X=X.
- An identity element with respect to designated by 1: X.1=1.X=X
- Commutative with respect to +: X=Y=Y+X.
- Commutative with respect to . : X.Y=Y.X.
- distributive over +: X.(Y+Z)-X.Y+X.Z
- distributive over: X+(Y.Z)=(X+Y).(X+Z)’
- For each element x belonging to B, there exist an element x’ or x called the complement of x such that x. x’=0 and x+ x’=1
- There exists at least two elements x, y belonging to B such that x#y these postulates were formulated by
**E.V Huntintong**in 1904 A.D and also known as**Huntintong****postulates**.

#### Duality principle of Boolean algebra :

The duality principle of Boolean algebra states that,” Every algebric expression deducible from the postulates of Boolean algebra remains valid if the operators and identity elements are interchanged”. In a two valid Boolean algebra B are the identity element of the same 1 and 0. If the dual and algebric expression desired, we simply Interchange OR and AND operations as replace (1’s by zeros and o’s by one)

#### Boolean algebra formulas

**Table of Boolean algebra Law are discuss Below:**

1. | Law of identify | $A=A$ $\overline{A}=\overline{A}$ |

2. | commutative law | $\mathrm{A}\xb7\mathrm{B}=\mathrm{B}\xb7\mathrm{A}$ A+B=B+A |

3. | Associative Law | $\mathrm{A}\xb7(\mathrm{B}\xb7\mathrm{C})=\mathrm{A}\xb7\mathrm{B}\xb7\mathrm{C}$ |

4. | Idempotent Law | A+(B+C)=A+B+C |

5. | Double Negative Law | $\mathrm{A}\xb7\mathrm{A}=\mathrm{A}$ A+A=A |

6. | Complementary Law | $\stackrel{-}{\stackrel{-}{A}}=A$ $A+\stackrel{-}{A}=1$ |

7. | Law of Intersection | $\mathrm{A}\xb71=\mathrm{A}$ $\mathrm{A}\xb70=0$ |

8. | Law of Union | $\mathrm{A}+1=1$ $\mathrm{A}+0=\mathrm{A}$ |

9. | DeMorgan’s Theorem | $\overline{AB}=\overline{A}+\overline{B}$ $\overline{\mathrm{A}+\mathrm{B}}=\overline{\mathrm{A}}+\overline{\mathrm{B}}$ |

10. | Distributive Law | $\mathrm{A}\xb7(\mathrm{B}+\mathrm{C})=(\mathrm{A}\xb7\mathrm{B})+(\mathrm{A}\xb7\mathrm{C})$ $\mathrm{A}\xb7\left(\mathrm{BC}\right)=(\mathrm{A}+\mathrm{B})+(\mathrm{A}+\mathrm{C})$ |

11. | Law of Absorption | $\mathrm{A}\xb7(\mathrm{A}+\mathrm{B})=\mathrm{A}$ $\mathrm{A}+\left(\mathrm{AB}\right)=\mathrm{A}$ |

12. | Law of common Identities | $\mathrm{A}\xb7(\overline{\mathrm{A}}+\mathrm{B})=\mathrm{AB}$ $\mathrm{A}+\left(\overline{\mathrm{A}}\mathrm{B}\right)=\mathrm{A}+\mathrm{B}$ |

**Simplify Boolean Expression using Boolean Algebra**

Using the Laws given above** Simplify Boolean Algebra Example are as follows in PDF**