Standard Representation for Logic Function

Logical functions are expressed in terms of different combinations of logical variables with their true forms as well as the complement forms. A logic function can be represented in the following forms:

• Standard Form
• Canonical Form

Standard Form

The standard form of the Boolean function is when it is expressed in sum of the products or product of the sums.
So, the two standard forms are:

1. Sum Of Product(SOP)  
2. Product Of Sum(POS)

Sum Of Product(SOP):

A SOP (Sum of Products) expression is the logical OR of product terms. That means, first logical ANDing of one or more literals and then ORing the products.

Given below is the logical function in SOP Form:

Y = AB + B'C + C'A  

Product Of Sum(POS):

The logical product of two or more logical sum terms is called a product of sums expression. That means, it is a boolean expression containing OR terms, called sum terms and the product denotes ANDing of sum terms.

Given below is a function in POS Form:

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

Canonical Forms

Canonical form is useful to specify Boolean functions in a form that allows comparison for equality and has a correspondence to the truth tables. Canonical Forms that are commonly used are: 

1.  Sum of Minterms (SOM) 
2.  Product of Maxterms (POM)

Sum Of Minterms:

It is just as the sum of products, here just the product terms are called minterms.

A product term containing all n variables of the function in either true or complemented form is called the minterm.These product terms are called the fundamental products or standard products or minterms. In the minterm, a variable will possess the value 1 if it is in true or uncomplemented form, whereas, it contains the value 0 if it is in complemented form. 

For a two-variable function, four different combinations are possible, A′B′, A′B, AB′, and AB. 

For a three-variable function, eight different combinations are possible, A'B'C', A'B'C, A'BC', A'BC, AB'C', AB'C, ABC', ABC.

Any boolean function can be expressed as sum of its minterms as:

F(list of variables) = Σ(list of min term indices)

Product Of Maxterms:

I   it is just as the product of sums, here just the product terms are called maxterms.

    A sum term containing all n variables of the function in either true or complemented form is called the maxterm. Each maxterm is obtained by an OR operation of the variables in their true form or complemented form. In the maxterm, a variable will possess the value 0 if it is in true or uncomplemented form, whereas, it contains the value 1 if it is in complemented form.These sum terms are called the standard sums or maxterms.

      Four different combinations are possible for a two-variable function, such as, A+B, A+B', A'+B, and A'+B'. 

      Eight different combinations are possible for a eight-variable function, A+B+C, A+B+C', A+B'+C, A+B'+C', A'+B+C, A'+B+C', A'+B'+C, A'+B'+C'. 

      Any boolean function can be expressed as sum of its maxterms as:

F(list of variables) =  Π(list of max term indices)

Obtain the canonical product of the sum form of the following function. F (A, B, C) = (A + B′) (B + C) (A + C′). 

In the above three-variable expression, C is missing from the first term, A is missing from the second term, and B is missing from the third term. 

Therefore, CC′ is to be added with first term, AA′ is to be added with the second, and BB′ is to be added with the third term. This is shown below.

 F (A, B, C) = (A + B′) (B + C) (A + C′) 

                    = (A + B′ + 0) (B + C + 0) (A + C′ + 0) 

                    = (A + B′ + CC′) (B + C + AA′) (A + C′ + BB′) 

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

 [using the distributive property, as X + YZ = (X + Y)(X + Z)]

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

[as (A + B′ + C′) (A + B′ + C′) = A + B′ + C′] 

Hence the canonical product of the sum expression for the given function is F (A, B, C) = (A + B′ + C) (A + B′ + C′) (A + B + C) (A′ + B + C) (A + B + C′) 

Obtain the canonical product of the sum form of the following function. F (A, B, C) = A + B′C 

In the above three-variable expression, the function is given at sum of the product form. 

First, the function needs to be changed to product of the sum form by applying the distributive law as shown below. 

F (A, B, C) = A + B′C = (A + B′) (A + C)

 Now, in the above expression, C is missing from the first term and B is missing from the second term. Hence CC′ is to be added with the first term and BB′ is to be added with the second term as shown below.

 F (A, B, C) = (A + B′) (A + C) 

                    = (A + B′ + CC′) (A + C + BB′)

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

 [using the distributive property, as X + YZ = (X + Y) (X + Z)]