Simplifying equations using boolean algebra

In summary, the conversation discusses simplifying equations using boolean algebra and suggests using standard identities such as associativity, commutativity, and distributivity. The conversation also mentions using trial and error and checking the truth table for accuracy. Finally, it provides an example solution for the first problem and encourages checking the solution for correctness.
  • #1
kirti.1127
10
0
1. Simplify the following equations using boolean algebra

2. a) abc + ab'c'+ ab'c

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


3. Please help me to solve the above equations
 
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  • #2
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?
 
  • #3
reply

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.
 
  • #4
kirti.1127 said:
ya. In second problem first i used sop and pos rules but got stuck up on next step.

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.
 
  • #5
reply

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

Ya i tried.
 
  • #6
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 expressionof course, sometimes it may be difficult to tell whether you have reduced your expression as simple as possible.
 
  • #7
another hint:
x.y + y = y
since
x.y+y = (x+1).y = (1).y = y
 
  • #8
reply

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
 
  • #9
kirti.1127 said:
abc+ab'c'+ab'c
=abc+a(b'+c')+ab'c (Product of Sum rule)

?? 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.
 
  • #10
Oh you u r correct.
that's y i told u to check.
 

What is boolean algebra?

Boolean algebra is a mathematical system used to simplify and manipulate logical expressions. It is based on the concept of binary values, where variables can be either true (represented by 1) or false (represented by 0).

Why do we use boolean algebra?

Boolean algebra is primarily used in digital electronics and computer science to design and analyze digital circuits. It can also be applied in other fields such as computer programming, logic, and mathematics.

What are the basic operations in boolean algebra?

The basic operations in boolean algebra are AND, OR, and NOT. AND represents the logical conjunction (denoted by & or *), OR represents the logical disjunction (denoted by + or |), and NOT represents the logical negation (denoted by ~ or ¬).

How do we simplify equations using boolean algebra?

To simplify equations using boolean algebra, we use the properties and laws of boolean algebra, such as commutative, associative, distributive, and De Morgan's laws. We also use truth tables and logical equivalences to determine the simplest form of an expression.

Can boolean algebra be applied to more than two variables?

Yes, boolean algebra can be applied to any number of variables. However, as the number of variables increases, the complexity of the expressions also increases, making simplification more challenging.

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