# boolean algebra simplification help~

#### minakrn

Joined Feb 4, 2007
3
hi guys ^^;

well.. i am new here and would greatly appreciate some help with the following algebra simplification problems (i have included my work so that i may compare the logic with you guys)
:

1) AC + ABC + BC

A + C + ABC + BC -> De'Morgans
A + ABC + BC + C -> Commutative
A(1 + BC) + BC + C -> Distributive
..stuck~

2) (A+B)(A + B)

(A * B)(A + B) -> De'Morgans
(A * A * B) + (A * B * B) -> Distributive
[simplified answer]? A + B -> Identity

3) ABC + AC

C(A * B + A) -> Distributive
C((A + A)(A + B)) -> Distributive
[simplified answer]? C * (A + B) -> Identity

4) BC + B(AD + CD)

BC + BAD + BCD -> Distributive
B(C + AD + CD) -> Distributive
..stuck~

5) (B + C + BC)(BC + AB + AC)

I basically tried to distribute everything.
Cancelled using identities and got stuck~

6) XYZ + XYZ + XYZ + XYZ

XZ + XYZ + XYZ -> Adjacency
XZ + X(YZ + YZ) -> Distributive
..stuck~ [Question: Can you simplify (YZ + YZ) to 1? (using the identity much similiar to X + X = 1)

7) XYZ + XYZ + XYZ + XYZ

XYZ + XYZ + XY -> Adjacency
Z(XY + XY) + XY -> Distributive
..stuck~ [Question: Can you simplify (YZ + YZ) to 1? (using the identity much similiar to X + X = 1)

Wow..that was tedious~

Well..there you go ^^;

Best regards~

#### Dave

Joined Nov 17, 2003
6,969
Hi and Welcome to All About Circuits!

Can I ask for one point of clarification? When you write for example AC are you meaning [ NOT(A) AND NOT(C) ] or are you meaning [ NOT(A AND C) ].

Thanks.

Dave

#### minakrn

Joined Feb 4, 2007
3
Hi Dave and thanks ^^;

Well..what I meant was:

[ NOT(A) AND NOT(C) ] -> so it looks more like this A C.

BTW: can you tell me what the difference is? Is it because the De'Morgans concept?

I apologize for this confusion.~

#### Dave

Joined Nov 17, 2003
6,969
To understand the difference, think about the truth tables:

A | NOT(A) | B | NOT(B) | A.B | NOT(A.B) | NOT(A) AND NOT(B)
--------------------------------------------------------------
0 | 1 | 0 | 1 | 0 | 1 | 1
0 | 1 | 1 | 0 | 0 | 1 | 0
1 | 0 | 0 | 1 | 0 | 1 | 0
1 | 0 | 1 | 0 | 1 | 0 | 0

Hence NOT(A.B) is a NAND (NOT-AND) gate, whereas (NOT(A) AND NOT(B)) is a NOR (NOT-OR) gate.

Clearer? EDIT: Apologies, the truth table hasn't come out as I expected, I recommend you write it down from the above layout but properly format the columns!

I will have a look through your initial workings when I get a moment later.

Dave