# What is boolean Algebraic laws and theorems? | What is Boolean algebra theorem?

## What is Boolean  Algebraic laws and theorems?

Boolean theorems and laws are used to simplify the various logical expressions. In a digital designing problem, a unique logical expression is evolved from the truth table. If this logical expression is simplified the designing becomes easier. The boolean algebra is mainly used in digital electronics, set theory and digital electronics.

### Commutative Law

X • Y = Y • X  and  X + Y = Y + X

### Associative Law

X + (Y + Z) = (X + Y) + Z  = X + Y + Z

And X • (Y • Z) = (X • Y) • Z  = X • Y • Z

### Distributive Law

1.X • (Y + Z) = X Y + X Z

2. (X + Y) • (W + Z) = X W+ X Z + YW + YZ

## Boolean Algebraic theorems

1. (X’)’ = X

X • 1 = X

X • X = X

X • X = 0

X • 0 = 0

X + 1 = 1

X + 0 = X

X + X = X

X + X = 1

### What is Boolean Algebra explain the Demorgan’s Theorem?

De Morgan’s theorem allows large bars in a Boolean Expression to be broken up into smaller bars over individual variables.

De Morgan’s theorem says that a large bar over several variables can be broken between the variables if the sign between the variables is changed.

De Morgan’s theorem can be used to prove that a NAND gate is equal to an OR gate with inverted inputs.

De Morgan’s theorem can be used to prove that a NOR gate is equal to an AND gate with inverted inputs.

In order to reduce expressions with large bars, the bars must first be broken up. This means that in some cases, the first step in reducing an expression is to use De Morgan’s theorem.

It is highly recommended to place parentheses around terms where lines have been broken.

1. (X • Y)’ = X ‘+ Y’

2. (X + Y )’= X’ • Y’

## Transposition theorems statement

(X + Y ) ( X + Z ) =  X + YZ

### Distributiin theorems

X + YZ = (X + Y ) ( X + Z )

### Redundancy Law

1. (X + Y) • (X + Y) = X

2. X Y + X Y = X

X • (X + Y) = X

X + X Y = X

### XOR Gate

X ⊕ Y = (X + Y) • (X + Y)  = X Y + X Y

### XNOR Gate

X ⊙ Y = (X + Y) • (X + Y)

### XNOR Gate

X ⊙ Y = (X + Y) • (X • Y)  = X Y + X Y

### Consensus theorem

• Used to eleminate redundant term.

• It is applicable only when if a boolean function,

1. Contains 3-variables

2. Each variable used 2-times

3. Only one variable is in complemented or uncomplemented form.

4. Then the related terms to that complemented or uncomplemented variable is the answer.

## Duality theorem

• “Dual expression” is equivalent to write a negative logic of the given boolean relation. For this we,

1. Changes each OR sign by an AND sign and vice-versa.

2. Complement any ‘0’ or ‘1’ appearing in expression.

3. Keep literals as it is.

• For 1-times Dual, it is called “Self Dual Expression”.

• For N-variables, maximum possible Self-Dual Function

(2)^2^n-1 = 2^(2^n/2).

• With N-variables, maximum possible distinct logic functions = 2^2^n

## Complementary theorem

For obtaining complement expression we

1. Changes each OR sign by AND sign and vice-versa.

2. Complement any ‘0’ or ‘1’ appearing in expression.

3. Complement the individual literals.