Conversion between Canonical FormsIn 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:
Conversion of POS to SOP formFor 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:
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 formFor 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:
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 formFor getting the standard SOP form of the given nonstandard 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.
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 nonstandard 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 formFor getting the standard POS form of the given nonstandard 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 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)
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