I have this given problem:
Expression Rule(s) Used
1. (A + C)(AD + AD') + AC + C Original Expression
2. (A + C)A(D + D') + AC + C Distributive.
3. (A + C)A + AC + C Complement, Identity.
4. A((A + C) + C) + C Commutative, Distributive.
5. A(A + C) + C Associative, Idempotent.
6. AA + AC + C Distributive.
7. A + (A + T)C Idempotent, Identity, Distributive.
8. A + C Identity, twice.
I'm really confused from step 4 to step 5. It tells me the rules used, but I'm not seeing where the extra +C went.
Also, I'm confused from step 6 to 8. How did we get A+T? And then it goes away in the next step. Again, it does tell me the rules, but I'm not seeing exactly how it is done.
Question: Can someone explain to me in better detail what's happening on steps 4->5 and 6->8
From 4 to 5, the $(A + C) + C$ term is simplified to just $A + C$ as follows:
By Associative, $(A +C) +C = A + (C + C)$
By Idempotence, $C + C=C$, and thus $A + (C +C) = A +C$
From 6 to 8: the $AC + C$ term gets simplified to just $C$ as follows:
$C =TC$ by Identity (the $T$ is the True constant ... many books use a $1$ instead)
and thus:
$AC + C = AC +TC$
By Distributive, this is $(A +T)C$
By Identity, $A +T=T$, and thus we get $TC$
But then we use the $TC=C$ Identity again, and we get $C$