CANONICAL AND
STANDARD FORMS
These are the special
cases for SOP and POS forms. They are :
a.
Standard SOP form.
b.
Standard POS form.
→STANDARD SOP FORM
:-
1. If each term of SOP form consists of all
the literals then the SOP form is known as Standard or Canonical SOP form.
2. Each individual term in standard SOP
form is known as “minterm”.
3.Therefore, the canonical SOP form is
also known as “minterm canonical form”.
For Example :-
F(P,Q,R)=PQR+P`QR`+PQR`.
→STANDARD
POS FORM:-
1. If each term in the POS form contains
all the literals then the POS form is known as Standard or Canonical POS form.
2. Each individual term is standard POS
form is known as “maxterm”.
For Example:-
F(P,Q,R)=(P+Q+R)(P`+Q+R`)(P+Q+R`).
·
Converting
expression in standard SOP :
1. SOP form can be converted into standard
SOP by ANDing the term expression by Oring the variable in complement and
uncompliment form.
2. For example, if there are three literals
in Boolean function for eg A,B,C and the term is AB.
3. Here the missing variable is C.
Therefore, we have to AND the variable C by ORing (C+C`) to term Ab+AB(C+C`).
Steps to convert SOP to Standard SOP
form
:
1. Find the missing literal in each product
term and we have to ANDing the literal by ORing the missing literal.
2.Expand the terms by applying
distributive law and reorder the literals in the product term.
3. Reduce the expression by omitting the
repeated product terms.
For Example :-
Convert the given expression to standard SOP form.
a. F(A,B,C)=AB+BC+AC
=
AB(C+C`)+BC(A+A`)+AC(B+B`)
=ABC+ABC`+BCA+BCA`+ACB+ACB`
=ABC+ABC`+ABC+A`BC+ABC+AB`C
=ABC+ABC`+A`BC+A`BC+AB`C.
b. F(A,B,C)=A+ABC
=A(B+B`)(C+C`)+ABC
=(AB+AB`)(C+C`)+ABC
=ABC+ABC`+ABC+AB`C+AB`C`
=ABC+ABC`+AB`C+AB`C`.
c. F(P,Q,R)=PQ+PR+QR
=PQ(R+R`)+PR(Q+Q`)+QR(P+P`)
=PQR+PQR`+PRQ+PRQ`+QRP+QRP`
=PQR+PQR`+PQR+PQ`R+PQR+P`QR
=PQR+PQR`+PQ`R+P`QR.
Steps
to convert POS to Standard POS form :
1. Finding the missing literal in each sum
term.
2. Now, we have to OR the missing term by
ANDing the literal in normal form and compliment form.
3. Expand the expression by applying
distributive law and reorder the literals in sum term.
4. Reduce the expression by omitting the
repeated sum terms.
For Example:-
Convert
POS form TO Standard POS form
a. F(A,B,C)=(A+B)(B+C)(A+C)
= [A+B+(C∙C`)][B+C+(A∙A`)][A+C+(B∙B`)]
=
(A+B+C)(A+B+C`)(B+C+A)(B+C+A`)(A+C+B)(A+C+B`)
=
(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).
b. F(A,B,C)=A+AB+BC
= A∙(B+B`)(C+C`)+AB∙(C+C`)+BC∙(A+A`)
= (AB+AB`)(C+C`)+ABC+ABC`+BCA+BCA`
= ABC+ABC`+AB`C+AB`C`+A`BC.
c. F(P,Q,R)=PQ+R+PR
= PQ(R+R`)+R(P+P`)(Q+Q`)+PR(Q+Q`)
= PQR+PQR`+PQR+PQR`+P`QR+P`Q`R+PQR+PQ`R
= PQR+PQR`+P`QR+P`Q`R+PQ`R.
d. F(X,Y,Z)=(XY+Z)(XZ+Y)
= XY∙XZ+XY∙Y+Z∙XZ+Z∙Y
= XYZ+XY+XZ+YZ
= XYZ+XY(Z+Z`)+XZ(Y+Y`)+YZ(X+X`)
= XYZ+XYZ+XYZ`+XYZ+XY`Z+XYZ+X`YZ
= XYZ+XYZ`+XY`Z+X`YZ.
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