Sum of Products and Product of Sums

The different forms of canonical expression which includes the sum of products (SOP) and products of the sum (POS), The canonical expression can be defined as a Boolean expression which has either min term otherwise max term. For example, if we have two variables namely X & Y then the canonical expression comprising of min terms will be XY+X’Y’, whereas the canonical expression comprising of max terms will be (X+Y) (X’+Y’). This article discusses an overview of Sum of Products and Product of Sums, types of SOP and POS, schematic design, and K-map.

Sum of Products and Product of Sums

The concept of the sum of products (SOP) mainly includes minterm, types of SOP, K-map, and schematic design of SOP. Similarly, the product of sums (POS) mainly includes the max term, types of product of sums, k-map and schematic design of POS.

What is a Sum of Product (SOP)?

The short form of the sum of the product is SOP, and it is one kind of Boolean algebra expression. In this, the different product inputs are being added together. The product of inputs is Boolean logical AND whereas the sum or addition is Boolean logical OR. Before going to understand the concept of the sum of products, we have to know the concept of minterm.

The min term can be defined as, when the minimum combinations of inputs are high then the output will be high. The best example of this is AND gate, so we can say that min terms are combinations of AND gate inputs. The truth table of the min term is shown below.

X

Y Z

Min Term (m)

0

0

0

X’Y’Z’ = m0

0

0 1

X’Y’Z = m1

0

1 0 X’Y Z’ = m2
0 1 1

X’YZ = m3

1 0 0

XY’Z’= m4

1

0 1 XY’Z = m5
1 1 0

XYZ’ = m6

1 1 1

XYZ = m7

In the above table, there are three inputs namely X, Y, Z and the combinations of these inputs are 8. Every combination has a minterm that is specified with m.

Types of Sum of Product (SOP)

The sum of products is available in three different forms which include the following.

  • Canonical Sum of Products
  • Non-Canonical Sum of Products
  • Minimal Sum of Products

1). Canonical Sum of Products

This is a normal form of SOP, and it can be formed with grouping the minterms of the function for which the o/p is high or true, and it is also called as the sum of minterms. The expression of the canonical SOP is denoted with sign summation (∑), and the minterms in the bracket are taken when the output is true. The truth table of the canonical sum of the product is shown below.

X

Y Z

F

0

0 0 0
0 0 1

1

0

1 0 1
0 1 1

1

1

0 0 0
1 0 1

1

1

1 0 0
1 1 1

0

For the above table, the canonical SOP form can be written as F = ∑ (m1, m2, m3, m5)
By expanding the above summation we can get the following function.
                                                                   F = m1 + m2 + m3 + m5
By substituting the minterms in the above equation we can get the below expression
                                                        F = X’Y’Z + X’YZ’ + X’YZ + XY’Z
The product term of the canonical form includes both complemented and non-complimented inputs

2). Non-Canonical Sum of Products

In the non-canonical sum of product form, the product terms are simplified. For example, let’s take the above canonical expression.
                                                          F = X’Y’Z + X’YZ’ + X’YZ + XY’Z
                                                             F = X’Y’Z + X’Y (Z’+Z) + XY’Z
Here Z’+Z =1 (Standard function)
                                                           F = X’Y’Z + X’Y (1) + XY’Z
                                                              F = X’Y’Z + X’Y + XY’Z
This is still in the form of SOP, but it is the non-canonical form

3). Minimal Sum of Products

This is the most simplified expression of the sum of the product, and It is also a type of non-canonical. This type of can is made simplified with the Boolean algebraic theorems although it is simply done by using K-map (Karnaugh map).

This form is chosen due to the number of input lines & gates are used in this is minimum. It is profitably useful due to its solid size, quick speed, along with low manufacture price.

Let’s take an example of canonical form function, and the minimal Sum of Products K map is

SOP K-map
SOP K-map

The expression of this based on the K-map will be

F = Y’Z + X’Y

Schematic Design of Sum of Product

The expression of the sum of product executes two-level AND-OR design, and this design requires a collection of AND gates and one OR gate. Each expression of the sum of the product has similar designing.

Schematic Design of SOP
Schematic Design of SOP

The number of inputs and the number of AND gates depend upon the expression one is implementing. The design for a minimal sum of product & canonical expression using AND-OR gates is shown above.

What is a Product of Sum (POS)?

The short form of the product of the sum is POS, and it is one kind of Boolean algebra expression. In this, it is a form in which products of the dissimilar sum of inputs are taken, which are not arithmetic result & sum although they are logical Boolean AND & OR correspondingly. Before going to understand the concept of the product of the sum, we have to know the concept of the max term.

The maxterm can be defined as a term that is true for the highest number of input combinations otherwise that is false for single input combinations. Because OR gate also provides false for just one input combination. Thus Max term is OR of any complemented otherwise non-complemented inputs.

X

Y Z Max Term (M)

0

0 0

X+Y+Z = M0

0 0 1

X+Y+Z’ = M1

0

1 0 X+Y’+ Z = M2
0 1 1

X+Y’+Z’ = M3

1

0 0 X’+Y+Z= M4
1 0 1

X’+Y+Z’ = M5

1

1 0 X’+Y’+Z = M6
1 1 1

X’+Y’+Z’ = M7

In the above table, there are three inputs namely X, Y, Z and the combinations of these inputs are 8. Every combination has a max term that is specified with M.

In max term, every input is complemented as it provides only ‘0’ while the stated combination is applied & complement of minterm is a max term.
                                                                                                            M3 =m3’
                                                                                                           (X’YZ)’ = M3
                                                                                    X+Y’+Z’=M3 (De Morgan’s Law)

Types of Product of Sums (POS)

The product of the sum is classified into three types which include the following.

  • Canonical Product of Sums
  • Non – Canonical Product of Sums
  • Minimal Product of Sums

1). Canonical Product of Sum

The canonical POS is also named as a product of max term. These are AND jointly for which o/p is low or false. The expression this is denoted by ∏ and the max terms in the bracket are taken when the output is false. The truth table of the canonical product of sum is shown below.

X

Y Z F
0 0 0

0

0

0 1 1
0 1 0

1

0

1 1 1
1 0 0

0

1 0 1

1

1

1 0 0
1 1 1

0

For the above table, the canonical POS can be written as F = ∏ (M0, M4, M6, M7)
By expanding the above equation we can get the following function.
                                                                                                     F = M0, M4, M6, M7
By substituting the max terms in the above equation we can get the below expression
                                                                            F = (X+Y+Z) (X’+Y+Z)(X’+Y’+Z)(X’+Y’+Z’)
The product term of the canonical form includes both complemented and non-complimented inputs

2). Non – Canonical Product of Sum

The expression of the product of sum (POS) is not in normal form is named as non-canonical form. For example, let’s take the above expression
                                                                                         F = (X+Y+Z) (X’+Y+Z)(X’+Y’+Z)(X’+Y’+Z’)
                                                                                                              F = (Y+Z) (X’+Y+Z) (X’+Y’+Z’)
Similar although reversed terms remove from two Max terms & forms only term to show it here is an instance.
                                                                                    = (X+Y+Z) (X’+Y+Z)
                                                                                    = XX’+XY+XZ+X’Y+YY+YZ+X’Z+YZ+ZZ
                                                                                    = 0+XY+XZ+X’Y+YY+YZ+X’Z+YZ+Z
                                                                                    = X (Y+Z) + X’ (Y+Z) + Y(1+Z) +Z
                                                                                    = (Y+Z) (X+X’) + Y (1) +Z
                                                                                    = (Y+Z) (0) +Y+Z
                                                                                     = Y+Z
The above final expression is still in the form of Product of Sum; however, it is in the form of non-canonical.

3). Minimal Product of Sums

This is the most simplified expression of the product of the sum, and it is also a type of non-canonical. This type of can is made simplified with the Boolean algebraic theorems although it is simply done by using K-map (Karnaugh map).

This form is chosen due to the number of input lines & gates are used in this is minimum. It is profitably useful due to its solid size, quick speed, along with low manufacture price.

Let’s take an example of canonical form function, and the Product of sums K map  is

POS K-map
POS K-map

The expression of this based on the K-map will be

                                                                                   F = (Y+Z) (X’+Y’)

Schematic Design of Product of Sum

The expression of the product of the sum executes two levels OR- AND design and this design requires a collection of OR gates and one AND gate. Each expression of the product of the sum has similar designing.

Schematic Design of POS
Schematic Design of POS

The number of inputs and the number of AND gates depend upon the expression one is implementing. The design for a minimal sum of product & canonical expression using OR-AND gates is shown above.

Thus, this is all about Canonical Forms: Sum of Products and Product of Sums, schematic design, K-map, etc. From the above information finally, we can conclude that a Boolean expression consists completely any of minterm otherwise maxterm is named as the canonical expression. Here is a question for you, what are the two forms of canonical expressions?

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