Search code examples
boolean-algebra

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!

Solution

  • 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