Simplifying equations using boolean algebra

kirti.1127

[b]1. Simplify the following equations using boolean algebra

[b]2. a) abc + ab'c'+ ab'c

b) (abc)'+(a+c)'+b'c'


[b]3. Please help me to solve the above equations

mjsd

pls first clarify your notations.
I suspect abc means a AND b AND c?
a+b means a OR b?
a' means NOT a?
right?

now: what standard identities do u know?

kirti.1127

ya. I could not use the complement notation while posting the query. I know all the standard notations. In second problem first i used sop and pos rules but got stuck up on next step. So please if u can help me in that.

mjsd

sop? pos?
what's that? anyway,

have u tried playing around using the standard axioms:
eg. associativity, commutativity, distributivity, De Morgan's Law, idempotence...etc.

for example:
a.a = a, a+a=a, (a.b).c = a.(b.c), a+b =b+a, a.(b+c)=a.b+a.c, a+(b.c)=(a+b).(a+c)
a+a'=1, a.a'=0
etc.

kirti.1127

sop-sum of product rule
and pos means product of sum rule.

Ya i tried.

mjsd

try reversing the distributive law to gather "common factor"
repeat use of axioms etc. it is a bit of a trial and error process, unless you can see something in advance (which comes with experience only). But you can always check, at each step, that you have not make an error by checking the truth table for both the original and derived expression.
another hint, sometimes it may even be useful to "add terms" into your expression


of course, sometimes it may be difficult to tell whether you have reduced your expression as simple as possible.

mjsd

another hint:
x.y + y = y
since
x.y+y = (x+1).y = (1).y = y

kirti.1127

I tried solving the first pb:
solution:

abc+ab'c'+ab'c
=abc+a(b'+c')+ab'c (Product of Sum rule)
=abc+ab'+ac'+ab'c
=abc+ac'+ab'(1+c)
=abc+ac'+ab' (1+c=1)
=abc+a(c'+b')
=abc+ab'c' (Sum of Product rule)

Just check if the solution is correct

mjsd

?? b'c' => b'+c' ?

is it b'c' or (b.c)' ?

you can always check answer by simply writing out the truth table for the original expression and then compare with the one for the new expression.

kirti.1127

Oh ya u r correct.
that's y i told u to check.