Understanding Boolean Functions: Sum of Products and Product of Sums

Hey friends, welcome to the YouTube  channel ALL ABOUT ELECTRONICS. So in this video, we will see that, how to  represent any Boolean function or the Boolean expression in a sum of product  and the product of sum form.

And we will also understand the  concept of minterms and the maxterms. So we know that in the digital electronics,  if we have any logic circuit, then its output is the function of the digital inputs.

And the relationship between the input and  output can be represented either using the truth table or using the Boolean expression. So from the truth table, we can understand  that, for what input combinations, the output

of the logic circuit is high. And the same thing can also be  represented using the Boolean expression. Now this Boolean expression or the Boolean  function can be represented either in the

sum of product form or in the product of sum form. So first, let's see the sum of  product form of representation. So in this SOP form, the product term is the  logical AND operation of the different input

variables. Where the variables could be in  the true form or in the complemented form. So these are the few examples of the product term. So in this A.B, both the  variables are in the true form.

While in this second case, this A and C are in  the true form, while the B is in the complemented form Similarly, in the third case, this A is in the  true form, while the B is in the complemented form.

Now in this SOP, this sum refers  to the logical OR operation. So in this sum of product form of expression,  different product terms are logically ORed. And this logical OR operation is  represented using this plus sign.

While these terms are the product terms. So let's see some expressions, which  are in the sum of product form. So if you see this function F1, then  it is in the sum of product form.

Because here, this A can be represented as A.A. So if you see all these three terms,  then they are the product terms. And this plus sign represents  the logical OR operation.

Similarly, let's say, this F2 is the function  of variable A and B, and this function is also in the SOP form. Now the expression in this SOP form can be  in the canonical form or in the non-canonical

form. So in this non-canonical SOP form, each product  term may or may not contain all the variables of the function.

For example, this first expression  is in the non-canonical form. So if you see this expression F1, then it  is the function of the three variables. But if you see this first product term,  then it contains only one variable.

While if you see this second and third  term, then they contain only two variables. That means each product term does not  contain all the variables of the function. Therefore, this expression  is in the non-canonical form.

On the other hand, in the canonical SOP form,  each product term contains all the variables of the function. Where the variables in each  product term can be in the true form or in the complemented form.

So this second expression  is in the canonical form. Because here, each product term contains  all the variables of the function. Similarly, if you see this function F3,  then it is also in the canonical SOP form.

Because here, this F3 is the  function of three variables. And here, each product term contains  all the variables of the function. Therefore, this expression  is in the canonical SOP form.

So in short, the sum of product form of  expression can be categorized in the two types. That is canonical SOP form and  the non-canonical SOP form. Similarly, now let's see the second type of  representation, which is known as the product

of sum. So as I mentioned earlier, this sum refers to  the logical OR operation, while this product refers to the logical AND operation.

So in this POS form of representation, this  sum term is the logical OR operation of the different variables. Where each variable could  be in the true form or in the complemented form.

So these are the few examples of the sum terms. So in this product of sum, or the POS form  of representation, different sum terms are logically ANDed.

So this dot represents the product or the  logical AND operation, while this term inside the bracket represents the sum terms. So that is the product of  sum form of representation.

So now let's see a few more expressions,  which are in the product of sum form. So here, this function F1 and F2 represents  the Boolean expression in the POS form. So as you can see, each expression contains  the sum terms, and these terms are logically

ANDed. Now similar to the SOP form, this product  of sum form of expression can be categorized in two types, that is canonical and non-canonical.

So in non-canonical type of POS form, each sum  term may or may not contain all the variables of the function. For example, this F1 and F2 are  non-canonical form of expressions.

Because if you see this F1, then it  is the function of three variables. But if you see the first sum term,  then it contains only two variables. Similarly, this last term is  also missing the variable A.

Similarly, if you see this F2, then  it is the function of four variables. But if you see each sum term, then it  does not contain all four variables. Therefore, these two expressions  are in non-canonical POS form.

On the other hand, in the canonical POS form  of expression, each sum term contains all the variables of the function. So if you see this function F3, then  it is in the canonical POS form.

Because here, each sum term contains  all the variables of the function. So these variables could be in the  true form or in the complemented form. So similarly, if you see these functions F4  and F5, then they are also in the canonical

POS form. So in short, this product of  sum form of representation   can be of two types, that is canonical and

the non-canonical. Now, if we just talk about the canonical form  of representation, then in the canonical SOP form, each product term is also called minterm.

And therefore, this canonical SOP form of  representation is also said to be a sum of minterms. Similarly, in the canonical  POS form of representation,  

each sum term is also called maxterm. And this canonical POS form of representation  is also said to be a product of maxterms. So first, let's understand what  is this minterm and the maxterm.

And let's start with the minterm. So this minterm is the product term which  consists of all the variables of the function either in the true form or  in the complemented form.

So now let's see, if we have some function  of n variables, then what are the possible minterms of that function. And if we have been given the truth table  of some logic circuit, then how we can write

its Boolean expression in the form of minterms. So if we have some function with two variables  A and B, then there are total four different possible input combinations.

And following are the minterms  corresponding to each combination. So now let's see how to write these minterms. So in the particular combination,  if the value of that variable is 0,  

then in the minterm expression, it can  be written in the complemented form. And if the value of that variable is 1, then  in the min term expression, it is written in the true form.

For example, in this first case, the  value of both variables A and B is 0. And therefore, they are written  in the complemented form. While in the second case, this  A is 0, while this B is 1.

Therefore, this A is written in the complemented  form, while this B is written in the true form. Similarly, in the third case,  this minterm is equal to AB bar.

And in the fourth case, this  minterm is equal to A.B. So that is how we can write all the  minterms according to the input combination. Now for convenience, these minterms are also  represented by the lower case letter m, followed

by the number, which is the decimal  equivalent of the input combination. That means this 0 0 corresponds to  m0, while this 0 1 corresponds to m1. Likewise, this 1 0 corresponds to  m2, and this 1 1 corresponds to m3.

Similarly, for the three variables,  the following are the minterms. Now since the min terms are the product terms,  so its output is 1, only for the specific input combination.

For example, this min term A bar dot b B bar  dot C bar is 1, when all the inputs are 0. Likewise, this minterm m3 is 1, when the  input A is equal to 0, and the input B and C are 1.

And likewise, the output of each minterm is  1, only for the specific input combination. So as you can see, for the three input  variables, there are total 8 minterms. Or in general, for the n input variables,  we have total 2 to the power n min terms.

So now let us see, if we have been given the  truth table of some logic circuit, then how to write its Boolean expression  in the form of minterms. So as you can see, this F1 is the  function of three variables A, B and C.

And this function F1 is 1, only for  the specific input combinations. Now if we want to write this function F1 in  the algebraic form, then what we will do, we will write down all the minterms corresponding  to those combinations for which the output

of this function is equal to 1. And then we will logically OR all these terms. And if you see this function F1,  then it is in the canonical SOP form.

Or we can say that it is the sum of minterms. Alternatively, we can also write this  expression as m0 plus m2 plus m4 plus m7. Because as we have seen, this A bar. B bar. C  bar is equal to m0, while this A bar. B .C bar is

equal to m0. While this A bar. B. C bar is equal to m2. Likewise, this A.B.C corresponds to m7,  while this A .B bar .C bar corresponds to m4.

That means alternatively, we can  also write this expression like this. And further in the more abbreviated  form, it can also be written like this. So here, this Σ represents  the summation of min terms.

And these decimal numbers are  the subscript of these minterms. So all these three expressions of  F1 are in the canonical SOP form. So similarly, let's take one more example.

So here, this function F2 is 1 only  for the three input combinations. So first, let's write down the minterms  corresponding to those input combinations. And then, let's perform the logical  OR operation of these minterms.

So further, this function F2   can be written in this form, where  this m1 is equal to A bar. B bar. C, while this m4 is equal to A .B bar. C bar.

And this m6 corresponds to A. B. C bar. And in the more abbreviated form,  it can also be written like this. So in this way, from the truth table, we can  write the Boolean expression of the function

in the form of minterms. Similarly, now let's see what is maxterm. So if we have a function with n variables,  then its maxterm is the sum term which contains

all the variables of the function, where  the variables could be in the true form or in the complemented form. So for a function with n different variables,  there are 2 to the power n different possible

maxterms. So let's take the case of two variables. So for the two variables A and B, there are  total four different input combinations.

And here are the maxterms  corresponding to each combination. So as you can see, each maxterm is the  summation of all the variables of the function. Now these variables can be in the  true form or in the complemented form.

And that depends on the value of the variable. So if the value of that variable is 0, then  in the maxterm, that variable is represented in the true form.

On the other hand, if the value of that variable  is 1, then in the maxterm, it is represented in the complemented form. So as you can see, in the first case,  the value of both variables A and B is 0.

And therefore, they are  represented in the true form. Similarly, in the second case, this A  is equal to 0, while B is equal to 1. So this A is represented in the true form,  while this B is represented in the complemented

form. And the same procedure is followed  for the remaining combinations. And for the convenience, these maxterms are  also represented by the uppercase letter M,

followed by the number, which is the  decimal equivalent of the input combination. So in this case, we have  maxterms starting from M0 to M3. Similarly, for three variables A, B  and C, following are the 8 maxterms.

So now let's see, if we have a truth table  of some digital circuit, then algebraically, how to write its output in the form of maxterms. So here is the truth table of the function F1.

And as you can see, the output of this function  is 1 for the four different combinations. So first, let's find the F1 bar. Now since the F1 bar is the complement of F1,  so when F1 is equal to 0, this F1 bar will

be equal to 1. So first of all, let's represent this  F1 bar in the form of sum of minterms. And for that, let's do the summation of all  the minterms for which this F1 bar is equal

to 1. So here, this minterm corresponding to 001 is  equal to A bar. B bar. C, while the minterm corresponding to 011 is equal to A bar. B. C

Likewise, the minterm corresponding to 101 is equal to A. B bar. C. And the minterm corresponding to  110 is equal to A. B. C bar. Now if we take the complement of this  F1 bar, then we will get the F1, right?

So using the De Morgan's law, let's simplify it. So just break this bar and give  it to the individual group. And at the same time, also change the sign.

So once again, using the De Morgan's law,  we will get the following expression, which is in the product of sum form. And here, each term is the maxterm.

Now if you closely observe, then this maxterm A+B+C bar corresponds to 001. Likewise, this maxterm A + B bar + C bar corresponds to 011. Similarly, this A bar + B + C bar corresponds to 101.

And likewise, this A bar + B bar +C corresponds to 110. So all these four combinations correspond  to maxterms M1, M3, M5 and M6. So in a simplified form, we  can also write it like this.

Or equivalently, it can also be written like  this. Where this π represents the product or the logical AND operation of all maxterms. And the number inside the bracket  represents the corresponding maxterms.

Now one more thing if you observe, then for  all these max terms, this function F1 is equal to 0. So directly from the truth table, if we want  to write this expression in the POS form, or

in the form of product of maxterm, then we  just need to consider the terms for which this function F1 is equal to 0. So let's take one more example,  so that it will get clear to you.

So here is the truth table of the function F2. So if we want to write it in the form of max terms, then just consider the terms for which this function F2 is equal to 0.

So these maxterms are 0, 1, 6 and 7. Or equivalently, it can also be written like this. And if we want to represent it in the algebraic  form, then we can also write it in this fashion.

So in this way, directly from the truth table,  we can write the function in the canonical POS form or in the product of maxterm. Alright, so for the three inputs A, B and  C, here are the corresponding minterms and

the maxterms. So if we see any maxterms, then its output  is equal to 0 only for the specific input combination.

For example, this A+ B+ C is equal  to 0, when all the inputs A, B and C are 0. Likewise, this A bar + B bar + C bar  is 0, when all the inputs A, B and C are 1. Moreover, if you observe, then each minterm  is the complement of the corresponding maxterm.

For example, if you take the complement of  this A + B + C, then we will get this A bar . B bar . C bar. Or we can say that, this M0 bar is equal to m0.

So in general, we can say that, these minterms are the complement of the maxterm. So using this, we can convert any canonical  SOP form into the equivalent POS form. And vice versa, we can convert any canonical  POS form into the equivalent SOP form.

So let's say, we have been given this function  F1 and it is in the canonical SOP form. And as you can see, it is the  function of three variables. Now if we take the complement of this F1,  then it contains all the minterms for which

this function F1 is equal to 0, right? That is equal to 1, 3, 5 and 6. Or we can write it as m1 + m3 + m5 + m6.

Now once again, if we take the complement  of this F1 bar, then we will get the F1. And using De Morgan's law, we can write it  as m1 bar . m3 bar . m5 bar . m6 bar. And as we have seen, these minterms  are the complement of the maxterm.

That means if we take the complement of these  minterms, then we will get the corresponding maxterms. That is equal to m1 . m3 . m5 . m6.

That means in the canonical POS form, this  expression can be written as m1 . m3 . m5 . m6. Or in a more abbreviated form,  it can also be written like this.

So in this way, we can convert any canonical  SOP form of expression into the equivalent POS form. And if you closely observe, then this POS  form contains the number which is not present

in the SOP form. That means without the truth table also, we  can convert any canonical SOP form into the equivalent POS form.

Or vice versa, we can convert any canonical  POS form into the equivalent SOP form. So to understand that,  let's take one more example. So let's say, we have given one  expression in the canonical SOP form.

And we want to find the equivalent POS form. So as you can see, this F2 is  the function of 3 variables. That means it will have total 8 minterms  and maxterms, starting from 0 to 7.

Now in the POS form, we will have the numbers  which are not present in the SOP form. So in this case, those numbers are 0, 3, 4 and 7. And therefore, this is the  equivalent POS form of expression.

So in this way, we can convert any one  canonical form into the other canonical form. So similarly in the next video, we will see  that, how to convert any non-canonical form into the equivalent canonical form.

So we will see that, how to convert the  non-canonical POS form of expression   into the equivalent canonical form.

And likewise, how to convert the non-canonical  SOP form of expression into the equivalent canonical form. But I hope in this video, you understood the  sum of product and the product of sum form

of representation of the Boolean expression. And you also understood the concept  of minterm and the maxterm. So if you have any question or suggestion,  then do let me know here in the comment section below

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