Boolean Algebra Video Lecture | Digital Electronics - Electrical Engineering (EE)

in this video we will understand about
boolean algebra basics so make sure you
watch this video completely to get your
basics right now the point number one is
fundamentals now here you have to
understand that 1 & 0 are not binary
they are not binary numbers ok so this
is wrong assumption the 1 and 0 are
logic 1 which means VCC and logic 0
which means ground this represents 1 & 0
here in this world now the variable a
can take both the values 1 & 0 so this
was one of the important point I want to
make sure now you can see the point 1 is
1 plus a is equal to 1 this means if you
are adding 1 plus 0 the answer is 1 or
you are adding 1 plus 1 the answer is
still 1 because it is not binary it is
logic 1 and logic 0 now the point number
2 is a plus a bar now a bar means
complement of the given variable so if a
is the variable a bar means the
complement of that variable so a plus a
bar the answer is 1 you can check by
yourself
1 plus 0 the answer is 1 0 plus 1 the
answer is still 1 now the point number 3
a into 0 the answer is 0 that's basic
then you have point number 4 a into a
bar that is equal to 0 because if is 1
then its complement is 0 if a is 0 then
its complement is 1 so 1 into 0 or 0
into 1 the answer is still 0 then you
have point number five a plus a the
answer is a now it's obvious 0 plus 0
answer is 0 1 plus 1 answer is 1 now the
point number 6 is a plus 0 the answer is
a it's obvious there is no explanation
needed for point number 6 then you have
0.7 which means a dot a is a that's
always again because if you are adding
multiplying 1 & 1
it's 1 if you are multiplying 0 & 0 the
answer is 0 this means a into a the
answer is a and the point number 8 it is
a into 1 the answer is a so this were
the important 8 points you should know
so in the previous slide we have studied
about the basics the fundamentals there
were total eight points now in this
slide we'll see
important results make sure you mark all
these five equations or you can say the
boolean expressions the point one is if
you have a plus a B you can take here a
common now if the bracket contains one
plus B now one plus anything the answer
is what the answer is one we have
understood that in the basics so one
plus anything the answer is one so a to
one the answer is e now these two points
you have understood because your basics
were clear now the expression is very
important you have a plus a B the answer
is what the answer is a now the second
point if you have a plus a bar B the
answer is equal to a plus B now in in
general you can say that X plus X bar Y
correct the answer is X plus one because
here instead of a plus a bar b the the
equation the expression can be a bar
plus a B or the expression can be a bar
plus a B bar so that is the reason I
have forgiven you this general
expression now you can sale answer is
what for this part the X is a bar and Y
is B correct and this term is X bar
which means a bar the whole bar the
answer is correct
so a bar plus a B the answer for that is
X plus y which means a bar plus B
similarly you can do it for this
expression a bar plus a B bar the x
value here is a bar the Y value here is
P bar so the answer is a bar plus B bar
so now we will see the point number
three a plus B into a plus C the answer
is a plus BC now it's simple you can
prove that how it's it's all about
multiplication and knowing your basics
right so if you have a plus B into a
plus C you have to do multiplication a
would be multiplied to this bracket
correct so a into a plus n to C then you
have B into a plus B into C now a into a
the answer is a and if you take a common
okay I will write whole expression again
and now from this part if you take a
common one plus anything answer is what
answer is one so this part is one so a
plus B C is your final answer now let us
move forward to the fourth expression
now this expression
because of the de Morgan's law a plus B
the whole bar the answer is a bar into
beaver
similarly the fifth point it's because
of the de Morgan's second law it is a
dot B this means a into B into B the
whole bar the answer is a bar plus B bar
in this slide we will understand about
consensus here now this expression is
pretty simple a B plus a bar C plus BC
if you have this then this term is
redundant and the answer is a B plus a
bar C and I will tell you how this term
became redundant and what is the proof
behind this equation it's simple you
have three variable in the left hand
side in the left hand side you have
three variables you have to go from
three variable to two variable correct
and how to do that it's simple it's
manipulation you have to manipulate your
LHS and then you can arrive at your RHS
so look carefully here I have done the
manipulation in this line a B plus a bar
C + VC into 1 now this term BC into 1 is
still BC correct now this one can be
written as e + e bar and then you will
get your answer because once you do this
one has a plus B bar now you can say the
term here is ABC and the term here is a
bar BC now we have to Club the common
part a B and ABC Club that then you have
a bar C then a bar CB Club that now here
you can take a B common the bracket
would be 1 + C and in this part you can
take a bar C common and the bracket
would be 1 plus B so 1 plus anything the
answer is 1 that is the reason you are
getting the answer as a B plus a bar C
now we will see important concept for
Gate point of view it is tool of boolean
expression in this part you have to do
the replacement if you see addition it
is replaced to multiplication and if you
see multiplication it would be replaced
to addition let us understand this by an
example a B plus a bar C plus BC is
equal to a B plus a bar C tell the dual
form of this expression actually this
expression is Consensus theorem now you
have to find the dual of this expression
how to go for the dual of this
expression wherever you see
go for the multiplication that is here
it is dot product okay and whenever you
see this dot which means multiplication
go for the addition so you can see here
a dot B since you're there is
multiplication you have to change that
multiplication into addition so the
answer would be a plus B then you have
this plus among this two term correct
so plus would be changed to
multiplication again multiplication
change to addition addition change to
multiplication multiplication change to
addition similarly do for the right-hand
side simple so this is the dual form
this expression is the dual form of the
given equation so the last part of the
basics and in the further slides we will
see the gate questions or based on the
boolean algebra basics cool let us start
property of boolean algebra the first
property is commutative property now
it's simple commutative property you can
see you have studied in this this kind
of property in probability as well a
plus B is equal to B plus a similarly a
into P is equal to P into a it won't
make any difference if you change the
order correct first a then B or first B
then a similarly goes for the third
equation into B the whole bar is equal
to B into ad whole part whereas the
order matters in matrix okay the order
matters in matrix but not in this
equations where a and B are treated as
variable okay whenever you have variable
order never matters but whenever you
have matrix then order will matter okay
so this was the important point now let
us move forward for the next property
associative property associative
property is just changing of the bracket
which term will you solve first if you
saw a plus B the whole bracket this
means you are solving this part first
then the answer will be added to plus C
and if you are doing a plus B plus C
bracket this means this part will be
solved first and then would be added to
it the answer will still be same so
bracket changing will also not matter
okay so here you can see a dot B the
whole bracket into C that is equal to a
to be dot see the whole bracket now the
third property is distributive property
a into B plus C a would be distributed
to every term so the answer is a B plus
AC similarly if you have a plus PC then
again a would be distributed to be a
would be distributed to C so a plus B
into a plus C so the fourth property is
transposition property this basically
means changing of the position so here
you can see a B plus a bar C B is with a
and C is with a bar so change their
proper change their position so in the
right hand side you can say a is with C
now and a bar is with B so the answer is
a plus C into a bar plus B now let us
move forward to the fifth point that is
literals now literals basically means
number of variables number of type of
variables that can be present in a
function so if you have a function which
contains the variables a B so the type
of variables that can be present in that
function is a a bar and BB bar so total
number of variables is 4 if you have
three here ABC so number of type of
variables that is literals how many
possible a a bar BB bar CC bar so n here
is 6 so in sum well you can say the
literals are 2n where n is the number of
variables of function simple so here the
N was 2 and that is why the literal was
2 into 2 which is 4 as you can see here
similarly here the N was 3 and the
literal was 2 into 3 which is 6 so you
can see here let's say this is capital n
which means the total number of literals
so I hope you got all the basics based
on boolean algebra now in the next video
we will see the gate questions based on
the basics of boolean algebra
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