Examples of Canonical Forms, Digital Electronics, CSE, GATE Video Lecture - Computer Science Engineering (CSE)

now let's practice some questions on
canonical forms let us say these are the
functions F 1 F 2 and F 3 and all the
functions are you know a functions of a
and B and C which means input is going
to be a B and C so now how can you write
F 1 so one thing is you can represent F
1 else sum of products if you are going
to represent their final sum of products
you have to take the min terms min terms
are nothing but the you know min terms
you know all the terms should be present
in it and now the min term should be
corresponding to the ones so if you want
to represent a fun you should pick this
one this one this one this one and this
one right so let me write them you the
numbers this is 0 this is 1 this is 2
this is 3 4 5 6 7 right so what are all
the values of f1 for which you know F 1
is 1 so what are all the values of the
function for which f 1 is 1 1 is 2 and
the other is 3 and the other is 5 6 and
7 so this is how if you write it in
canonical collection of you know this
sorry in this product of sum operates
form another way to write it is product
of sums form right so the same function
you could represent it like pi PI of how
can you write it whatever terms are not
here you put it them you put them there
so what are the terms not there
0 1 then 2 3 is there for is not there 5
it's 7 right that is so you could write
and the next one is f to practice F 2
also so what are the how can I write F 2
F 2 can be written as Sigma which means
sum up know our sum of products what are
the products that we have formed now it
is 0 2 4 6 right and how can you write
products product of sum is 1 3 5 7 right
and what about F 3 how when you write F
3 F 3 is Sigma what are the sum 0 1 6 7
and similarly PI whatever is missing
there
two three four five right so this is how
you could have present every function so
given the table given through table you
can convert them into functions like
this right and other one is instead of
giving you the function these two table
the function then use the expression now
they will use some expression which is
either a product of sum or sum of
product expression which is not a
canonical form not in canonical formulas
the terms which are forming the
expression might not be min terms or max
terms then what you should do is convert
them to canonical forms let us see how
to do that okay now given this
expression not the function how can you
convert into canonical form either this
product of sum or sum of products
whatever it is you have to convert it or
let us know kind of constrain or
construct it into this sum of products
now I want canonical sum of products
right so which means I want the min
terms now or it is a sum of products but
they do not canonical some operates why
because every term should be a min term
which means in every term all the
variable should be present but if you
look at it all the variables are not
present here right so there are two ways
to solve it one way is let the
expression be like this and keep on
modifying the expression until we get
the canonical form that is one way other
waves take this expression and convert
into table how am in that table the
function right so you write what are all
the possibilities for XYZ and then you
try to evaluate it and write it there
and wherever you have seen once you put
a Sigma and you get it two ways I mean
what I am trying to say is you could
have even you could even construct a
table like this and it contains XYZ
values and here you can write x prime y
plus Z prime plus X 5/3 and now for
every combination you find out the value
let us say the values are 0 1 0 1 like
this right then wherever you have 1 then
you can put a Sigma there and that is
nothing but canonical sum of products
and wherever there is 0 you can put a
CPI there and it is nothing but product
of sums that is one way other way is you
can take this expression as it is and
you can keep on expanding it until
everything is included
right I'll just tell you how to do it
see now it is X prime Y is there now if
this term has to be a min term what is
the variable that is missing Z right so
if you had a fired add Z then how can I
add it is the property is anything when
multiplied having multiplication means
here ending anything when handed with
one is nothing but same thing right so
what I write is like this X prime while
that is missing for that I am writing 1
plus Z prime X and y are missing
therefore I am writing 1 and 1 plus
there is nothing missing in this so
leave it as it is X 5 red now here one
is that is missing therefore for this
one I am going to put the Z terms how
can I write Z in terms of one this way
see this X prime Y Z plus Z prime plus
here X plus x prime y plus y prime into
Z prime plus X Y Z right now check this
now what is it X prime Y Z Y I am
combining these plus X prime Y Z prime
plus ok one thing you must be getting
note is we are always trying to minimize
the expression minimize the function or
the expression but here if you look at
it I am actually taking a smaller
expression and I am expanding it a
bigger one right why do you think that
we are doing this see whenever we are
trying to design a circuit right from
scratch then we are going to take the
expression and minimize it so that this
circuit will be minimal right in the
other other sense if already circuit is
existing and if you are going to make
use of the existing circuit then they
will give you standard interface
standard interface means the interface
will contain the inputs will contain all
the minterms
if you are going to use the existing I
know circuits in order to form the new
circuit
then your input should contain all the
minterms therefore if you are using this
standard model standard the available
infrastructure to create the new
functions then obviously we have to you
know have all the minterms
if you are trying to design something
from the scratch and if you want to
minimize the circuit then we apply the
minimization so this is actually a
reverse method right see previously we
have taken big terms and converted to
small terms now we are take small
demanding a big one right so do not get
confused both methods are useful and
important okay and now let us look at
this what are the terms here so here if
I take this one this one and this one is
going to be X Y Z prime plus if I take
this one this one and this one it is
going to be X Y prime Z prime plus this
X is over right now X prime X prime by Z
prime plus which means this one right X
prime Y Z prime plus X prime Y prime Z
prime X prime Y prime Z prime plus and
the remaining is XYZ XYZ right so now
there are most of if you look at it I
think many of the terms will be
repeating so one two three four five six
seven so we have seven terms actually
there are eight min terms anyway we got
seven min terms let's see if any of them
are repeating if you cannot see it this
way let me write the binary
representation of this I mean decimal
representation of these each of the min
term what is 0 1 1 3 what is 0 1 0 2
what is 1 1 0 what is 1 1 0 6 and what
is 1 0 0 4 what is 0 1 0 2 what is 0 0 0
it is 0 let's say X prime Y prime Z
prime is 0 0 0 it is 7 all right now
let's see if anything is repeating is
too is repeating therefore you need you
don't need it to test you can cross one
of them right so what is the final
answer you could have left it like this
it is nothing but the canonical law
no some of products you could leave it
like this or you can write it like this
in a shortcut so what is it it is better
you put in ascending order zero is there
- is there three is there for is there
six Izzy seven is there right it is
nothing but the canonical sum of
products now what is the canonical
product of sums if you have found it out
it means that for all these values the
function is 1 then what are the values
for which function is 0 it would put PI
right and what is remaining 0 1 is
remaining 2 3 is there for is there 5 5
is remaining right 1 comma 5 right you
please cross-check if I did everything
right or wrong I think it is right
I'll just verify it once yeah it is fine