# Boolean Algebraic Theorems

Boolean algebraic theorems are the theorems that are used to change the form of a boolean expression. Sometimes these theorems are used to minimize the terms of the expression and sometimes they are used just to transfer the expression from one form to another.

There are boolean algebraic theorems in digital logic:

1. De Morgan’s Theorem :
DE Morgan’s Theorem represents two of the most important rules of boolean algebra.

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

Thus, the complement of the product of variables is equal to the sum of their individual complements.

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

Thus, the complement of the sum of variables is equal to the product of their individual complements.

The above two laws can be extended for n variables as

```(A1 . A2 . A3 ... An)' = A1' + A2' + ... + An'

And

(A1 + A2 + ... + An)' = A1' . A2' . A3' ... An' ```

2. Transposition Theorem :
It states that:

`AB + A'C = (A + C) (A' + B)`

Proof:

```RHS
= (A + C) (A' + B)
= AA' + A'C + AB + CB
= 0 + A'C + AB + BC
= A'C + AB + BC(A + A')
= AB + ABC + A'C + A'BC
= AB + A'C
= LHS ```

3. Redundancy Theorem :
This theorem is used to eliminate the redundant terms. A variable is associated with some variable and its complement is associated with some other variable and the next term is formed by the left over variables, then the term becomes redundant.

Example:

`AB + BC' + AC = AC + BC' `

Proof:

```LHS
= AB + BC' + AC
= AB(C + C') + BC'(A + A') + AC(B + B')
= ABC + ABC' + ABC' + A'BC' + ABC + AB'c
= ABC + ABC' + A'BC' + AB'C
= AC(B + B') + BC'(A + A')
= AC + BC'
= RHS ```

4. Duality Theorem :
Dual expression is equivalent to write a negative logic of the given boolean relation. For this,

1. Change each OR sign by and AND sign and vice-versa.
2. Complement any 0 or 1 appearing in the expression.
3. Keep literals as it is.

Example:

`Dual of A(B+C) = AB + AC`

5. Complementary Theorem :
For obtaining complement expression,

1. Change each OR sign by AND sign and vice-versa.
2. Complement any 0 or 1 appearing in the expression.
3. Complement the individual literals.

Example:

`Complement of A(B+C) = A'B' + A'C'`

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