Boolean Expressions(SOP and POS forms)

Boolean Expressions(SOP and POS forms)

Any digital circuit can be expressed as Boolean Expression. Boolean Expression is mainly of two forms. These are Sum of Product (SOP) form and Product of Sum(POS) form.

Sum of Product (SOP) expression example:

           

            Z=(A’B’C’)+(A’B’C)+(A’BC’)

Product of Sum (POS) expression example:

                Z=(A’+B’+C).(A+B’+C).(A+B+C)

If we have n variables we can have 2ⁿ  terms associated with it i.e. if we have two variables we will have 4 combinations. Let’s understand with a table-

Variables		Minterm	Maxterm
A	B
0	0	A’B’	A+B
0	1	A’B	A+B’
1	0	AB’	A’+B
1	1	AB	A’+B’

Each minterm is obtained from an AND term of variables with each variable being primed , if corresponding bit of binary no is zero and unprimed if it is 1.

Each maxterm is obtained from an OR term of variables with each variable being unprimed,if corresponding bit is zero and primed if it is 1.Maxterm is complement of its corresponding minterm.
Example:
Express the following expression into sum of product and product of sum form-
f=(xy+z)(y+xz)
=xy.y+xy.xz+z.y+z.xz
=xy+xyz+yz+xz

X	Y	Z	F=xy+yz+xz+xyz
0	0	0	0	T0
0	0	1	0	T1
0	1	0	0	T2
0	1	1	1	t3
1	0	0	0	T4
1	0	1	1	t5
1	1	0	1	t6
1	1	1	1	t7

The corresponding minterm is where the f value is 1:
              f=xy+yz+xz+xyz=t3+t5+t6+t7=x'yz+xy'z+xyz'+xyz
                                                             =sum of minterms=sum of products

The corresponding maxterms is where the f value is 0:
             f=T0.T1.T2.T4
              =(x+y+z)(x+y+z')(x+y'+z)(x'+y+z)=product of sum

If you see closely that t'(i)=T(i)
Note that: The equation of above can also be represented as =∑(3,5,6,7)  or ∏(0,1,2,4)
Exercise:
Find the SOP and POS of f=∑(1,3,7) [for problem in finding email me]
Canonical Form:
If a function is either in sum of minterms or product of maxterms form,then the function is of canonical form.Standard Form:
If a function is in POS or SOP form it is called standard function.
If a function has equal no of minterms and maxterms than the function is called Neutral Function.



Digital Electronics Main Page
Truth Table

Karnaugh's Map

Combinational Circut

Sequential Circuit

Number System