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# Boolean Postulates and Theorem

## BOOLEAN POSTULATES and THEOREM

Boolean algebra was introduced by George Boole in 1854. Boolean algebra is used to perform binary and logical operations and differs in this respect from the ordinary algebra.

## Postulates of Boolean Algebra

1.    Identity Law :

A + 0   =          A

A.1      =          A

2. Commutative Law:

(A+B)        =          (B+A)

A.B            =          B.A

3. Associative Law:

A + (B + C)     =          (A + B) + C

A.(B.C)     =          (A.B).C

4. Distributive Law:

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

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

5. Complement Law:

A+A’         =          1

A.A’           =          0

## Boolean Algebra Theorems

1.    Duality Theorem: It states that we can derive a Boolean relation from another Boolean relation by just :

i.              Changing the ‘.’ With a ‘+’ and ‘+’ with a ‘.’

ii.            Complement the ‘0’s and ‘1’ in the expression

iii.           Leaving the variables unchanged

–       Examples:

–       A + A’ = 1;      Its dual will be           A . A’ = 0

2.    DeMorgan’s 1st Theorem: It states that “Complement of the products is equal to the sum of the complements

(A .B)’             =          A’ + B’

Or simply putting it another way a NAND gate can be replaced with a bubbled OR gate.

3.    DeMorgan’s 2nd Theorem: It states that the Complement of the sum terms equals the product of the complements.

( A + B)’          =          A’ . B’

Or putting it another way is “We can replace a NOR gate with a bubbled AND gate

4.    . Idempotency Theorem: The product or the sum of the same variable is the variable itself.

A+A = A

A . A = A

5.    Involution Theorem: Double complement of a variable is equal to the variable itself.

(A’)’     =          A

6.    Absorption Theorem:

A + (A.B)        =          A

A . (A + B)      =          A

7.    Consensus Theorem:

AB + A’C + BC          =          AB + A’C

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

8.    Uniting Theorem

AB + AB’        =          A

(A + B).(A + B’)          =          A

9.    Other Theorem

A + (A’ . B)     =          B

A . (A’ + B)     =          B

10. A + 1 =  1;      and     A.0 = 0

### Examples-1 Solve A.(AB + C)

Solution:

A.(AB + C)     =          AAB + AC                  (Using A.A = A)

=          AB + AC

=          A ( B + C)

### Example-2: A + A’B

Solution:

A + A’B           =          A + AB + A’B             (Using A+AB = A)

=          A + B(A+A’)

=          A + B

### Example-3: Solve Y = AB’D + AB’D’

Solution:

Y =      AB’D + AB’D’

= AB’ (D + D’)

= AB’

### Example-4 : Y= (AB’(C+BD) + A’B’)C

Solution:

Y         =          (AB’(C+BD) + A’B’)C

=          (AB’C + AB’BD + A’B’)C

=          (AB’C + A’B’)C

=          AB’CC + A’B’C

=          AB’C + A’B’C

=          B’C( A + A’)

=          B’C

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