Simplification of this Boolean expression

[momentum]

How can we write the below red marker boolean expression to simplify the equation?

Which Law is this?

B+!BA=B+A

Is this correct?

 

[tnich]

How can we write the below red marker boolean expression to simplify the equation?

Which Law is this?

B+!BA=B+A

View attachment 225715

Is this correct?

Try drawing a Venn diagram of $B+\bar BA$. That should show you how to answer your question.

 

[George Jones]

B+!BA=B+A

Are you asking about how to show this?

If so, @tnich gives one way. Here is a hint for another way. What is an equivalent expression for

$$X + YZ?$$

 

[EmilioL]

Theorem: B+(!BA) = B+A
Proof:
B+!BA = (B + !B) (B + A) (distribution of "or" over "and")
= 1 (B + A) (tautology: B or !B = 1)
= B + A (by the definiion of "and", 1 X = X)
Q.E.D.

BTW, this is a hideous notation. "=" in this context really means "if and only if". And 1 really means "true". The repurposing of integers 0 and 1 and of the arithmetic operators is more confusing than helpful--especially since in regular algebra, addition does not distribute over multiplication: 1 + (2 * 3) <> (1 + 2) (1+ 3). Boolean algebra is a red-headed stepchild of abstract algebra--but the meat and potatoes of formal logic. It's also a great place to learn to do rigorous proofs.

All the above steps can be shown to be valid via truth tables. Whether or not they are also formal "laws" depends on the axiom system you are using.
Boolean algebra is really propositional logic--a subject that can be developed informally via truth tables (really a model for the theory) or formally via an axiom system in various ways. For example, 0 can be a constant and 1 defined as "!0" -- or the reverse. The idea is to have axioms and at least one rule of inference so one doesn't have to keep resorting to truth tables (which is like counting on your fingers) in order to evaluate complex expressions. Then you can get juicy results like proving that your axiom system is consistent and (for propositional logic) complete...and you're off to the races!

I know this is massive overkill for an (old) homework question, but theorems and proofs are rather important in mathematics. :-)

Whenever someone would introduce the late, great John von Neumann as a "computer scientist", he would always say, "I am a mathmatician. There is no such thing as computer science--it's all technology."