How do i work this Boolean Algebra?

Question

Simplify: ~P~VSTC + ~PV~STC + ~PVS~TC + ~PVSTC + P~V~STC + P~VS~TC + P~VSTC + PV~S~TC + PV~S TC + PVS~TC + PVST~C + PVSTC (hint: ending value only has seven terms...) I have no clue how or why that's answer... If someone can please explain to me that would be sooooo great The answer is PVST + STC + VTC + VSC + PTC + PSC + PVC Thanks for the help guys... I got the answer from a calculator thing

Answer 1

Here are some general principles that will help:

Adjacency

$PQ+PQ'=P$

For example, the last two terms from your starting expression can be combined to form $PVST$. But you can also combine $PVSTC$ with $P'VSTC$ to form $VSTC$. Fortunately, you don't have to choose one, since you can always create a copy by:

Idempotence

$P=P+P$

Another useful one is:

Reduction

$P+P'Q= P +Q$ (the $P$ term 'reduces' the $P'Q$ term to $Q$)

and this one actually generalizes to:

Generalized Reduction

$PR+P'QR=PR+QR$

In your expression, for example, after obtaining $VSTC$, you can use that to reduce $P'V'STC$ to $P'STC$

Good luck!