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# Conversion between Canonical Forms

In our previous section, we learned about SOP(sum of product) and POS(product of sum) expressions and calculated POS and SOP forms for different Boolean functions. In this section, we will learn about how we can represent the POS form in the SOP form and SOP form in the POS form.

For converting the canonical expressions, we have to change the symbols ∏, ∑. These symbols are changed when we list out the index numbers of the equations. From the original form of the equation, these indices numbers are excluded. The SOP and POS forms of the boolean function are duals to each other.

There are the following steps using which we can easily convert the canonical forms of the equations:

1. Change the operational symbols used in the equation, such as ∑, ∏.
2. Use the Duality’s De-Morgan’s principal to write the indexes of the terms that are not presented in the given form of an equation or the index numbers of the Boolean function.

## Conversion of POS to SOP form

For getting the SOP form from the POS form, we have to change the symbol ∏ to ∑. After that, we write the numeric indexes of missing variables of the given Boolean function.

There are the following steps to convert the POS function F = Π x, y, z (2, 3, 5) = x y’ z’ + x y’ z + x y z’ into SOP form:

1. In the first step, we change the operational sign to Σ.
2. Next, we find the missing indexes of the terms, 000, 110, 001, 100, and 111.
3. Finally, we write the product form of the noted terms.

000 = x’ * y’ * z’

001 = x’ * y’ * z

100 = x * y’ * z’

110 = x * y* z’

111 = x * y * z

So the SOP form is:

F = Σ x, y, z (0, 1, 4, 6, 7) = (x’ * y’ * z’) + (x’ * y’ * z) + (x * y’ * z’) + (x * y* z’) + (x * y * z)

## Conversion of SOP form to POS form

For getting the POS form of the given SOP form expression, we will change the symbol ∏ to ∑. After that, we will write the numeric indexes of the variables which are missing in the boolean function.

There are the following steps used to convert the SOP function F = ∑ x, y, z (0, 2, 3, 5, 7) = x’ y’ z’ + z y’ z’ + x y’ z + xyz’ + xyz into POS:

• In the first step, we change the operational sign to ∏.
• We find the missing indexes of the terms, 001, 110, and 100.
• We write the sum form of the noted terms.

001 = (x + y + z)

100 = (x + y’ + z’)

110 = (x + y’ + z’)

So, the POS form is:

F = Π x, y, z (1, 4, 6) = (x + y + z) * (x + y’ + z’) * (x + y’ + z’)

## Conversion of SOP form to standard SOP form or Canonical SOP form

For getting the standard SOP form of the given non-standard SOP form, we will add all the variables in each product term which do not have all the variables. By using the Boolean algebraic law, (x + x’ = 0) and by following the below steps we can easily convert the normal SOP function into standard SOP form.

• Multiply each non-standard product term by the sum of its missing variable and its complement.
• Repeat step 1, until all resulting product terms contain all variables
• For each missing variable in the function, the number of product terms doubles.

Example:

Convert the non standard SOP function F = AB + A C + B C

Sol:

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

So, the standard SOP form of non-standard form is F = A B C + A B C’ + A B’ C + A’ B C

## Conversion of POS form to standard POS form or Canonical POS form

For getting the standard POS form of the given non-standard POS form, we will add all the variables in each product term that do not have all the variables. By using the Boolean algebraic law (x * x’ = 0) and by following the below steps, we can easily convert the normal POS function into a standard POS form.

• By adding each non-standard sum term to the product of its missing variable and its complement, which results in 2 sum terms
• Applying Boolean algebraic law, x + y z = (x + y) * (x + z)
• By repeating step 1, until all resulting sum terms contain all variables

By these three steps, we can convert the POS function into a standard POS function.

Example:

F = (p’ + q + r) * (q’ + r + s’) * (p + q’ + r’ + s)

1. Term (p’ + q + r)

As we can see that the variable s or s’ is missing in this term. So we add s*s’ = 1 in this term.

(p’ + q + r + s*s’) = (p’ + q + r + s) * (p’ + q + r + s’)

2. Term (q’ + r + s’)

Similarly, we add p*p’ = 1 in this term for getting the term containing all the variables.

(q’ + r + s’ + p*p’) = (p + q’ + r + s’) * (p’ + q’ + r + s’)

3. Term (q’ + r + s’)

Now, there is no need to add anything because all the variables are contained in this term.

So, the standard POS form equation of the function is

F = (p’ + q + r + s)* (p’ + q + r + s’)* (p + q’ + r + s’)* (p’ + q’ + r + s’) * (p + q’ + r’ + s)