Is "$p + p'q = p + q$" a law in Boolean Algebra?

Question

Is "$p + p'q = p + q$" a law in Boolean Algebra?

$$p.(q+r)=(p.q)+(p.r)\\p+(q.r)=(p+q).(p+r)\\p+\overline{p}\ q=p+q$$

My book says that it is one of the distributive laws, but I don't see it anywhere on the internet or any other books. Is it actually a distributive law or is my book incorrect?

Answer 1

We can use the distributive law to prove it. $$ p+q = p+1q = p+(p+p')q = p+pq+p'q = p(1+q)+p'q=p1+p'q=p+p'q $$

Answer 2

We can use the distributive law to prove it.

$$p+p'q = (p+p')(p+q) = 1(p+q) = p+q$$

Answer 3

$$p+q=p(q+q')+q(p+p')=pq+p'q+pq'=p+p'q=q+pq'$$

because $pq+pq=pq$ and $p+p'=1$.

Answer 4

Yes, it is a law, but you're right, very few sources list it ... which is too bad, since it's such a handy dandy rule!

Because it is such an uncommonly listed rule, I doubt you will find a high degree of consensus as far as names go even if you do see it listed, but I've seen it referred to as Reduction (of course, don't ask me to remember where I saw that :P ... and a quick Google search hasn't been very successful either :( ). But I really like the name Reduction (it's certainly how I call it myself): the $P$ term reduces the $P'Q$ term to just $Q$.

So while in $P + PQ$ the $P$ term absorbs the $PQ$, in $P + P'Q$, the $P$ term reduces term $P'Q$

Also, if you haven't ran into this one:

$PQ + PQ' = P$

That one is a bit more commonly listed again, and is fairly consistently known as Adjacency (because in a K-Map, the two terms that are being merged represent adjacent groups of cells that can be merged into a single grater block of cells represented by the simpler term) ... this is another super handy dandy rule ... but also often not listed.