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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

Phil
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1 Answers1

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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 $C$

Bram28
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