What is Sum Of Product (SOP) Form?

In Digital Electronics any logic circuit’s output is the function of digital inputs and the relation between input and output can be represented using logic table or boolean expressions.

This Boolean expression can be represented in two forms.

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

The Sum of Product Form

In the sum of the product form of representation, The product num is logical and operation of the different input variables where the variables could be in the true form or in the complemented form.

Example:

A.B, A.B̅.C (example of product term)

In SOP sum refers to logical OR Operation. Therefore, in this sum of product form of expression, we perform logical or operations on different product terms. Therefore it is known as the Sum of Product form.

Example:

A+ B̅.C + A.C

as we can see in above example the product terms(A , B̅.C , A.C) which are created by multiplying input variables are are summed with each other .

SOP Form

The SOP form can be in either canonical form or non-canonical form.

1. Non-Canonical SOP Form

In this form each product term between may or may not contain all the variables of the function.

Example:

F(A,B,C) = A+ B̅.C + A.C

as we can see in above example the function have variables A, B, C but we are not including each variable in each product term. in first product term (A) we have not included B & C. In second product term we (B̅.C) have not included A. While in last product term we have not included B.

2. Canonical SOP Form

In canonical SOP form each product term contains all the variables of the function, where variables in each product term can be in true form or complemented form.

Example :

F(A,B) =  Ā.B + A.B̅

As we can see in above example each product term contain all the variables which are present in function. In first one(Ā.B) A is present in complementary form while B is in true form. In second one A is present in true form while B is in complementary form.

Writing Boolean Expression for Given Truth Table in SOP Form

Suppose following is the truth table for an electronic circuit:

A	B	C	F
0	0	0	0
0	0	1	0
0	1	0	1
0	1	1	0
1	0	0	1
1	0	1	1
1	1	0	1
1	1	1	1

The SOP form is written when the function if high. i.e we will take conditions only in which the output is one. In given truth table the columns with F = 1 are the rows with high output. Now we will write function in the form of given variables which are A, B and C. If we have 1 in the cell of given variable we will write it as it is , and if we have 0 in the given cell then we will write it’s complement.

F = Ā.B.C̅ + A.B̅.C̅ + A.B̅.C + A.B.C

This is the POS in canonical form for the given truth table. Each expressions which lies in between the + are known as Minterms.

Example :

A.B.C, A.B̅.C

Therefore we can also write it as

F(A, B, C) = M2 + M4 + M5 + M6 + M7
F(A,B,C) = ∑m(2, 4, 5, 6, 7)

we can further reduce it to non-canonical SOP form by using boolean algebra.

F = Ā.B.C̅ + A.B̅.C̅ + A.B̅.C + A.B.C
F = Ā.B.C̅ + A.B̅.[C̅ + C] + A.B.[C̅ + C]
F = Ā.B.C̅ + A.B̅ + A.B
F = Ā.B.C̅ + A.[B̅ + B]
F = A + B.C̅

This is how we can get non-canonical SOP form of boolean expression.

Creating a Truth Table with SOP Boolean Expressions

Suppose following is the SOP expression for which we will make truth table now

F(A, B, C) = A'B' + AB + AC

1. As we have three input values (A, B, C) we will create 2^3 = 8 rows in truth table to include all input combinations.

A	B	C
0	0	0
0	0	1
0	1	0
0	1	1
1	0	0
1	0	1
1	1	0
1	1	1

2. for each input combination we will put values of input in expression and calculate the output:

F(A, B, C) = A'B' + AB + AC
F(0, 0, 0) = 0'0' + 00 + 00 = 0 + 0 + 0 = 0
F(0, 0, 1) = 0'0' + 00 + 01 = 0 + 0 + 0 = 0
F(0, 1, 0) = 0'1'  + 01 + 00 = 0 + 0 + 0 = 0
F(0, 1, 1) = 0'1' + 01 + 01 = 0 + 0 + 0 = 0
F(1, 0, 0) = 1'0' + 10 + 10 = 0 + 0 + 0 = 0
F(1, 0, 1) = 1'0' + 10 + 11 = 0 + 0 + 1 = 1
F(1, 1, 0) = 1'1' + 11 + 10 = 0 + 1 + 0 = 1
F(1, 1, 1) = 1'1' + 11 + 11 = 0 + 1 + 1 = 1

3. Create truth table form given input combinations and calculated output.

A	B	C	F
0	0	0	0
0	0	1	0
0	1	0	0
0	1	1	0
1	0	0	0
1	0	1	1
1	1	0	1
1	1	1	1

FAQs on Sum Of Product (SOP) Form

Q.1: What is difference between Canonical and Non-canonical form?

Answer:

This are two different ways to represent boolean expressions in Sum of Product (SOP) form. In canonical form , every product term contains all variables of the function and are included in either true or complementary form. On the other hand, in non-canonical form, product terms may or may not include all variables, and they may also can include all variables. For systematic representation canonical SOP form is used , while ono-canonical form is more concise.

Q.2: How to write Boolean expression in SOP form from truth table?

Answer:

• Take the rows with output 1 in given truth table.
• From the selected rows, convert the inputs in their true or complementary form.
• Write product terms for each product term. Use logical AND in between input variables.
• Combine all product terms using logical OR operation.

Q.3: How can I simplify a Boolean expression in the Sum of Products (SOP) form?

Answer:

We can use boolean algebra rules and law to simplify the equation

1. A + AB = A (Absorption Law)
2. A’BC + AB’C + ABC = A’BC + AB’C (Consensus Theorem )
3. AB + AC = A(B + C) (Distribution)
4. (A + B)’ = A’B’ (Demorgans Theorem)
5. Eliminate repeated terms , and the terms that have no impact on the function.
6. A + A’ = 1, AA’ = 0 (complement law)