CODEWITHSUNDEEP
boolean-algebra
Lorenzo Battilocchi
Lorenzo Battilocchi

Reputation: 1158

How to simplify using boolean algebra?

I have the following problem in my assignment:

"Verify, using Boolean algebra, the following equality:

NOT((A AND NOT B) OR (NOT A AND B)) == ((A AND B) OR (NOT A AND NOT B)). "

I am able to do it with Karnaugh Maps and Truth tables, but I'm stuck on the formal procedure using Boolean Algebra.

Thanks in advance for your kind help!

Upvotes: 0

Views: 230

Answers (2)

Lorenzo Battilocchi
Lorenzo Battilocchi

Reputation: 1158

I figured it out on my own:

Steps:

  1. ~((A AND ~B) AND (~A AND B)) .... Original Eqn.

  2. ((~A OR ~~B) AND (~~A OR ~B)) .... DeMorgan's Law

  3. ((~A OR B ) AND (A OR ~B) .... Elimination of double negation

Introducing mathematical symbols as it makes it a bit clearer in my opinion

  1. (~A (~B + A) * B (~B + A) .... "Factor out" (~A * B) and carry out multiplication

  2. (~A*~B)+(~AA)+(B~B)+(B*A) .... "Multiply out the terms"

  3. (~A*~B) + 1 + 1 + (B*A) .... Excluded middle

  4. (~A*~B) + (A*B) .... Required Answer

Upvotes: 1

Deepak
Deepak

Reputation: 61

Using DeMorgan's laws (http://www.ask-math.com/de-morgans-law.html) we can simplify down the left side:

  1. !((A and !B) or (!A and B))
  2. !(A and !B) and !(!A and B)
  3. (!A or B) and (A or !B)

Next we use the Product of Sums to get:

(!A and A) or (B and A) or (!A and !B) or (B and !B)

Since (!A and A) is false and (B and !B) is false we reduce to:

(B and A) or (!A and !B).

This matches the right side of the equation.

Upvotes: 3

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