Laws of Boolean Algebra

Laws of Boolean Algebra:

There are various Laws of Boolean Algebra, but we have only covered according to I.S.C. XII Syllabus.

Following are the laws which we will cover:

1	Property of "0"	A + 0 = A     A . 0 = 0
2	Property of "1"	A + 1 = 1     A . 1 = A
3	Idempotence Law	A + A = A     A . A = A
4	Involution Law	(A')' = A
5	Complementarity Law	A + A' = 1     A . A' = 0
6	Commutative Law	A + B = B + A     A . B = B . A
7	Associative Law	( A + B ) + C = A + ( B + C )     ( A . B ) . C = A . ( B . C )
8	De-Morgan's Law	( A + B )' = A' . B'     ( A . B )' = A' + B'
9	Distributive Law	A . ( B + C ) = A.B + A.C     A + ( B.C ) = ( A+B ). ( A+C )
10	Absorption Law	A + A.B = A     A . ( A + B ) = A
11	Redundancy Law	A + A'. B = A + B     A . ( A' + B ) = A + B

Let us discuss them one by one.

1)    Property of "0":

    This law states that if we perform AND/ OR of any input with 0 then the result will be as follows:

                1)    A + 0 = A

                2)    A . 0 = 0

    This can be expressed using following truth tables.

       1) A + 0 = A 

A	“0”	A+0
0	0	0
1	0	1

       

    2) A . 0 = 0 

A	“0”	A.0
0	0	0
1	0	0

2)    Property of "1":

    This law states that if we perform AND/ OR of any input with 0 then the result will be as follows:

                1)    A + 1 = 1

                2)    A . 1 = A

    This can be expressed using following truth tables.

       1) A + 1 = 1

A	“1”	A+1
0	1	1
1	1	1

       

    2) A . 1 = A 

A	“1”	A.0
0	1	0
1	1	1

3)    Idempotence Law:

        This law states that OR/ AND of a variable with itself is always equal to the variable.

            1)    A + A = A

            2)    A . A = A

The same can be expressed using following truth table:

A	A	A + A	A . A
0	0	0	0
1	1	1	1

4)    Involution Law:

        This law states that double complement of a variable is always equals to that variable.

                (A')' = A

    This can be understood by following truth table:

A	A’	(A’)’
0	1	0
1	0	1

5)    Complemetarity Law:

        Complementarity Law states 2 rules:

        Rule 1:    "OR" of a variable with its complement is always equal to 1

                        A + A' = 1

A	A’	A + A’
0	1	1
1	0	1

        Rule 2:    "AND" of a variable with its complement is always equal to 0

                        A . A' = 0

A	A’	A . A’
0	1	0
1	0	0

6)    Commutative Law:

This law states that the order of the variables connected using OR/ AND makes no

difference

                1)    A+B = B+A

                2)    A.B = B.A

    Below Truth Tables prove the above laws: 

 

    1)    A+B = B+A   

A	B	A + B	B + A
0	0	0	0
0	1	0	0
1	0	0	0
1	1	1	1

    

    2)    A.B = B.A

A	B	A . B	B . A
0	0	0	0
0	1	0	0
1	0	0	0
1	1	1	1

7)    Associative Law :

This law states that when more than two variables are connected using OR/ AND, the result is the same regardless of the grouping of the variables.

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

Below Truth Table illustrates these laws as applied to 3-input OR gates.

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

				(1)		(2)
A	B	C	B+C	A+(B+C)	A+B	(A+B)+C
0	0	0	0	0	0	0
0	0	1	1	1	0	1
0	1	0	1	1	1	1
0	1	1	1	1	1	1
1	0	0	0	1	1	1
1	0	1	1	1	1	1
1	1	0	1	1	1	1
1	1	1	1	1	1	1

        You can easily see that Col (1) & Col (2) have same values, thus verifying the Law.

            

            2)    A . ( B.C ) = ( A.B ).C

				(1)		(2)
A	B	C	B.C	A.(B.C)	A.B	(A.B).C
0	0	0	0	0	0	0
0	0	1	0	0	0	0
0	1	0	0	0	0	0
0	1	1	1	0	0	0
1	0	0	0	0	0	0
1	0	1	0	0	0	0
1	1	0	0	0	1	0
1	1	1	1	1	1	1

    here also Col (1) & Col (2) have same values, thus verifying the Law.

8)    De-Morgan's Law :

De-Morgan proposed two theorems.

            1)    (A + B)' = A' . B'

The complement of "OR" of two or more variables is equivalent to the "AND" of the complements of the individual variables.

            2)    (A . B)' = A' + B'

The complement of "AND" of two or more variables is equivalent to the OR of the complements of the individual variables.

    The same can be understand by following truth tables:

1)    (A + B)' = A' . B'					1	2
A	B	A'	B'	A.B	(A.B)'	A' + B'
0	0	1	1	0	1	1
0	1	1	0	0	1	1
1	0	0	1	0	1	1
1	1	0	0	1	0	0

2)    (A . B)' = A' + B'					1	2
A	B	A'	B'	A+B	(A+B)'	A' . B'
0	0	1	1	0	1	1
0	1	1	0	1	0	0
1	0	0	1	1	0	0
1	1	0	0	1	0	0

            In both the tables you can see that col. 1 = col. 2, hence proving the laws.

9)    Distributive Law:

This law states that for any given 3 or more variables, following relation establishes:

                1)    A(B + C) = AB + AC

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

Let us prove both the laws by using a truth tables;

1)    A(B + C) = AB + AC

				(1)			(2)
A	B	C	B+C	A.(B+C)	A.B	A.C	AB + AC
0	0	0	0	0	0	0	0
0	0	1	1	0	0	0	0
0	1	0	1	0	0	0	0
0	1	1	1	0	0	0	0
1	0	0	0	0	0	0	0
1	0	1	1	1	0	1	1
1	1	0	1	1	1	0	1
1	1	1	1	1	1	1	1

here col. 1 is equals to col.2, hence proving the law

2)    A + B.C = (A+B).(A+C)

				(1)			(2)
A	B	C	B.C	A+ B.C	A+B	A+C	(A+B). (A+C)
0	0	0	0	0	0	0	0
0	0	1	0	0	0	1	0
0	1	0	0	0	1	0	0
0	1	1	1	1	1	1	1
1	0	0	0	1	1	1	1
1	0	1	0	1	1	1	1
1	1	0	0	1	1	1	1
1	1	1	1	1	1	1	1

here also col. 1 is equals to col.2, thus verifying the Law.

10)    Absorption Law:

            This law states following 2 rules:

                1)    A + A . B = A

            

                This rule can be proved by applying the Distributive law & Property of 1 as follows:

                    A + AB 

                    = A( 1 + B)                   Factoring (distributive law)

                    = A . 1                                         (Property of 1)

                    = A                                             (Property of 1)

            The same can be seen in the Truth Table below :

A	B	A.B	A+( A.B)
0	0	0	0
0	1	0	0
1	0	0	1
1	1	1	1

                

                

                2)    A. ( A+B ) = A

                        

                This rule can be proved by applying the Distributive, Idempotence law & Property of 1 as follows:

                    A . (A+B)

                    = A . A + A . B          (distributive law)

                    = A + A . B                                 (Idempotence law : A.A = A)

                    = A( 1 + B)                                Factoring (distributive law)

                    = A . l                                         (Property of 1)

                    = A                                             (Property of 1)

            The same can be seen in the Truth Table below :

A	B	A+B	A.(A+B)
0	0	0	0
0	1	1	0
1	0	1	1
1	1	1	1

11)    Redundancy Law :

             This law states following 2 rules:

            1)    A+A'.B = A+B

            This rule can be proved as follows:

                    A + A'.B

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

                    = (1) . (A+B)                                 (Complementarity Law)                                                             = A + B                                         ( Property of 1) 

            

The same can be seen in the Truth Table below :

A	B	A’	A’. B	A+ B	A+ (A’. B)
0	0	1	0	0	0
0	1	1	1	1	1
1	0	0	0	1	1
1	1	0	0	1	1

            2)    A.(A'+B) = A . B

            This rule can be proved as follows:

                    A.(A'+B)

                    = A . A' + A . B                        (Distributive Law)

                    = 0 + A . B                               (Complementarity Law)

            The same can be seen in the Truth Table below :

A	B	A’	A’+ B	A.B	A. (A’+ B)
0	0	1	1	0	0
0	1	1	1	0	0
1	0	0	0	0	0
1	1	0	1	1	1