SOP and POS

Input Output Relation:

A	B	C	Y
0	0	0	0
0	0	1	1
0 : 1	1 : 1	0 : 1	1 : 1

Above Table1.. A,B,C are the inputs whereas Y represents the output of the system. The system may have a single output or multiple outputs.

The relation between inputs and output can be represented by :

1) Truth Table                                     2) Logic Diagram                    3) Switching Equations

1) Truth Table:

                   Here the relation between inputs and output can be represented by tabular form as shown in Table1. The state of output 0 or 1 is written for all possible input combinations.

2) Logic Diagram:

                     This is a diagrammatic way of expressing the input-output relationship of a digital circuit.           

3) Switching Equations:

                   The relation between inputs and output can be represented in the form of equation(s) called as switching equations.

                                   

                                    E.g. Y= A'BC + AB'C + ABC

                        The switching equations also called as Boolean equations. The switching equations or Boolean equations can be of two different types.

i) Sum of Products(SOP)                            ii) Product of Sum(POS)

Any Boolean function that is expressed as a sum of minterms is called as SOP(Sum of Product).

Any Boolean function that is expressed as a product of maxterms is said to be POS(Product of Sum).

It mainly involves in two Boolean terms, “minterms” and “maxterms”.

Min terms

A minterm is defined as the product term of n variables, in which each of the n variables will appear once either in its complemented or un-complemented form. The min term is denoted as mi where i is in the range of 0 ≤ i < 2ⁿ.

A variable is in complemented form, if its value is assigned to 0, and the variable is un-complimented form, if its value is assigned to 1.

Any Boolean function can be expressed as the sum (OR) of its 1- min terms. The representation of the equation will be

·         F(list of variables) = Σ(list of 1-min term indices)

        Ex: F (x, y, z) = Σm (3, 5, 6, 7)  i.e. Minterms 3,5,6,7 is giving output '1' for this function.

Max terms

A maxterm is defined as the product of n variables, within the range of 0 ≤ i < 2ⁿ. The max term is denoted as Mi. In maxterm, each variable is complemented, if its value is assigned to 1, and each variable is un-complemented if its value is assigned to 0.

Any Boolean function can be expressed the product (AND) of its 0 – max terms. The representation of the equation will be

·         F(list of variables) = Π (list of 0-max term indices)

          Ex: F (x, y, z) = ΠM (0, 1, 2, 4)     i.e. Minterms 0,1,2,4 is giving output '0' for this function.

The below table will make you understand about the representation of the mean terms and max terms of 3 variables.

Representation of Truth Table using Standard SOP Expression:

Rules:

1) For the given truth table, consider only those combinations of inputs which produce output=1.

2) Write down a product term interms of input variables for each combination.

3) OR all these product terms produced in step2 to get standard SOP.

A	B	Y
0	0	0
0	1	1
1	0	1
1	1	1

For the given truth table, follow above rules to get the resultant equation in SOP form i.e.

F(A,B)=∑m(1,2,3) = A’B+AB’+AB

Representation of Truth Table using Standard POS Expression:

Rules:

1) For the given truth table, consider only those combinations of inputs which produce output=0.

2) Write down a product term interms of input variables for each combination.

3) OR all these product terms produced in step2 to get standard POS.

A	B	Y
0	0	0
0	1	1
1	0	1
1	1	1

For the truth table, follow above rules to get the resultant equation in SOP form i.e.:

Y= πM(0)=A’B’

Conversion of SOP form to POS form

To convert the SOP form into POS form, first we should change the Σ to Π and then write the numeric indexes of missing variables of the given Boolean function.

Example:

The SOP function

F = ∑ A, B, C (0, 2, 3, 5, 7) = A’ B’ C’ + A B’ C’ + A B’ C + ABC’ + ABC

  

To convert it into POS follow the given steps.

Step 1: changing the operational sign to Π

Step 2: writing the missing indexes of the terms, 001(1), 100(4) and 110(6). Now write the sum form for these noted terms.

001 = (A + B + C’)              100 = (A’ + B + C)            110 = (A’ + B’ + C)

Writing down the new equation in the form of POS form,

F(A,B,C)=ΠM(1,4,6) = (A + B + C’) * (A’ + B + C) * (A’ + B’ + C)

Conversion of POS form to SOP form

To convert the POS form into SOP form, first we should change the Π to Σ and then write the numeric indexes of missing variables of the given Boolean function.

Ex: The POS function F = Π A, B, C (2, 3, 5) = A B’ C’ + A B’ C + ABC’ is written in SOP form by

Step 1: changing the operational sign to Σ

Step 2: writing the missing indexes of the terms, 000(0), 001(1), 100(4), 110(6), and 111(7). Now write the product form for these noted terms.

000 = A’ * B’ * C’                       001 = A’ * B’ * C                          100 = A * B’ * C’

110 = A * B* C’                                  111 = A * B * C

Writing down the new equation in the form of SOP form,

F(A,B,C)=Σm(0,1,4,6,7) = (A’ * B’ * C’) + (A’ * B’ * C) + (A * B’ * C’) + (A * B* C’) + (A * B * C)