## Basic Properties of Switching Algebra

**Basic Properties of Switching Algebra**

Switching algebra in some cases behaves slightly differently from the regular algebra. Let us take a look at some of the laws of the switching algebra which will help us derive some rules for simplification of the switching expressions.

**Idempotency Law of Switching Algebra**

The Idempotency law of switching Algebra states that

x + x = x

x.x = x

The method of perfect induction can be used to prove the Idempotency Law. In method of perfect induction, the theorem is verified for each and every possible value of the variable. Such method is valuable if the number of different values for which theorem is to be verified is small.

x + x = x stands verified since x can assume two different values 0 and 1 and for these values

0 + 0 = 0 and

1 + 1 = 1

Both are true , hence the equation is true.

The equation x.x = x stands verified since x can assume two different values 0 and 1 and for these values

0.0 = 0 and

1.1 = 1

Both are true , hence the equation is true.

**Commutative law of switching algebra**

The switching algebra holds the commutative law, which states that

x + y = y + x ;

x.y = y.x

The commutative law can be proved using the method of perfect induction

Associative law of switching Algebra

The convention for the parenthesizing in switching algebra is same as that of ordinary algebra. This means that a + b.c means a+(b.c). The associative law states that

(x+y) + z = x + (y+z) and

(x.y).z = x.(y.z)

**Complementation Law**

The Complementation law states that

x + x’ = 1

xx’ = 0

**Distributive Law**

The Distributive law of the multiplication is similar to the ordinary algebra. The multiplication distributes over the addition

x.(y+z) = x.z + x.z

It is the distribution of the addition over the multiplication that is different than the ordinary algebra. The following distribution law is also true in the switching algebra

x+y.z = (x+y).(x+z)

**Principal of Duality**

In all of the above equations we see that the equations stay valid if we interchange AND and OR. This is true for other equations as well. In other words if a equation is true, then it is true for its duel as well.

Simplification of Switching Algebraic Expressions

Some of the principles presented above can be used for simplification of the switching algebraic expression. By simplifying the algebraic expressions, the circuit required to realize the logic is minimized.

Some Helpful Rules for Minimization of switching functions

1. x+xy =x

Proof : x + xy = x.1+xy

= x.(1+y)

= x.1

= x

2. x.(x+y) = x

Proof : 1. By principle of duality – interchanging AND and OR.

2. x.(x+y) = x.x +x.y

= x+xy

=x

3. x+x’y= x+y

Proof : x+x’y = (x+x’)(x+y)

= 1(x+y)

= x+y

4. x.(x’+y) = x.y

Proof : 1. By principle of duality of equation 3 ( interchanging OR with AND and vice versa)

2. x.(x’+y) = x.x’ + x.y

= 0 + x.y

= x.y

5. xy+x’z+yz = xy+x’z

This theorem is also called consensus theorem and is helpful in the simplification of algebraic expressions.

Proof : xy+x’z+yz = xy+x’z+ yz.1

= xy+x’z+ yz.(x+x’)

= xy+x’z+ yz.x + yzx’

=xy(1+z) + x’z(1+y)

=xy.1 + x’z.1

= xy + x’z

6. This theorem is dual of the above consensus theorem

(x+y).(x’+z).(y+z) = (x+y)(x’z)