Can I use Idempotency Here?

Question

If I have statement like: AB + 'AC + BC, can I use Idepotency to remove BC and simplify, or does the AND between AB and 'AC rules this out?

More specifically to simplify:

AB+BC+C'A - Commutativty

AB+C+C'A - Idempotency?

AB+C'A - Idempotency?

Answer 1

Idempotence says that $P = P + P$ ... none of what you do is like that.

You can, however do this (this is assuming your $'AC$ is $A'C$):

$AB+BC+A'C \overset{Adjacency}= AB + ABC + A'BC + A'C \overset{Absorption \ x \ 2}= AB+ A'C$

Assuming your $'AC$ is $C'A$:

$AB + BC + C'A \overset{Adjacency}= ABC + ABC' + BC + C'A \overset{Absorption \ x \ 2}= BC + C'A$

This equivalence is known as the Consensus Theorem:

Consensus Theorem

$PQ + P'R + QR = PQ + P'R$

By the way, the Absorption I do can be done using Adjacency and Idempotence:

$AB +ABC \overset{Adjacency}= ABC + ABC' +ABC \overset{Idempotence} ABC +ABC' \overset{Adjacency}= AB$

... maybe that is what you were intuiting?