# Axioms and Laws of Boolean Algebra

In this tutorial, we are going to learn about the Axioms and Laws of Boolean Algebra in Digital Electronics.
Submitted by Saurabh Gupta, on November 16, 2019

Boolean Algebra differs from both general mathematical algebra and binary number systems. In Boolean Algebra, A+A =A and A.A = A, because the variable A has only logical value. It doesn't have any numerical significance. In ordinary mathematical algebra, A+A = 2A and A.A = A2, because the variable A has some numerical value here. Also, in Binary Number System 1+1 = 10, and in general mathematical algebra 1+1 = 2 but in Boolean Algebra 1+1 = 1 itself. Unlike ordinary algebra and Binary Number System here is subtraction or division in Boolean Algebra. We only use AOI (AND, OR and NOT/INVERT) logic operations to perform calculations in Boolean Algebra.

## Axioms in Boolean Algebra

There are some set of logical expressions which we accept as true and upon which we can build a set of useful theorems. These sets of logical expressions are known as Axioms or postulates of Boolean Algebra. An axiom is nothing more than the definition of three basic logic operations (AND, OR and NOT). All axioms defined in boolean algebra are the results of an operation that is performed by a logical gate.

```    Axiom 1: 0.0 = 0    Axiom 6: 0+1 = 1
Axiom 2: 0.1 = 0    Axiom 7: 1+0 = 1
Axiom 3: 1.0 = 0    Axiom 8: 1+1 = 1
Axiom 4: 1.1 = 1    Axiom 9: 0 = 1
Axiom 5: 0+0 = 0    Axiom 10: 1 = 0
```

Based on these axioms we can conclude many laws of Boolean Algebra which are listed below,

Commutative Laws

```    A+B = B+A, and
A.B = B.A
```

Associative Laws

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

AND Laws

```    A.0 = 0
A.1 = A
A.A = A
A.A = 0
```

OR Laws

```    A+0 = A
A+1 = 1
A+A = A
A+A = 0
```

Complementation Laws

```    If A = 0 then A = 1
If A = 1 then A = 0
A̿ = A
```

Distributive Laws

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

Idempotence Law

```    A.A = A, If A=1, then A.A = 1.1 =1 = A and if A=0, then A.A = 0.0 = 0 = A
A+A = A, If A=1, then A+A = 1+1 =1 = A and if A=0, then A+A = 0+0 = 0 = A
```

Absorption Law

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

De-Morgan's Law

```    A+B =  A. B
A.B = A + B
```

Consensus Theorem

```    A) AB + A C + BC = AB + A C
```

Proof:

```LHS     = AB + A C + BC
= AB + A C + BC (A+A)
= AB + A C + ABC + ABC
= AB (1+C) + A C (1+C)
= AB + A C = RHS
```
```    B) (A+B) (A + C) (B+C) = (A+B) (A + C)
```

Proof:

```LHS     = (A+B) (A + C) (B+C)
= (AA + AC + BA + BC) (B+C)
= (AC + BA + BC) (B+C)
= ABC + ACC + BAB + BAC + BCB + BCC
= ABC + AC + BA + BAC + BC + BC
= ABC + AC + BA + BAC + BC
= AC(B+1) + BA (1+C) + BC
= AC + BA + BC ............. (Equation 1)

RHS     = (A+B) (A + C)
= AA + AC + BA + BC
= AC + BA + BC ............. (Equation 2)
```

Since, Equation 1 = Equation 2, Hence Consensus Theorem is verified.