CODEWITHSUNDEEP
booleanboolean-logicboolean-expression
potatomeister
potatomeister

Reputation: 65

Boolean Algebra expression with nots

So at the end of my truth table I got:

NOT  NOT ABC.NOT ABD

________
___ ___
ABC.ABD

Does this simplify into ABC+ABD? Also what laws would I use?

Upvotes: 0

Views: 62

Answers (1)

elldur
elldur

Reputation: 2103

For a start I don't like the way they teach binary simplification. To make things much easier for yourself take these rules instead of the ones they taught you:

  1. A AND B = AB
  2. A AND A = A
  3. A OR B = A + B - AB
  4. NOT A = 1 - A

You can work out xor and etc.. yourself using these.

Now then using these rules:

NOT(NOT(A AND B AND C) AND (NOT (A AND B AND D)))

= 1 - ((1 - ABC)*(1 - ABD))

= 1 - (1 - ABC)(1 - ABD)

= 1 - 1 + ABD + ABC - ABCABD

= ABD + ABC - ABCD

= AB(D + C - CD)

= A AND B AND (C OR D)

which is equivalant to ABC + ABD using your system.

Edit: First 3 equals (but a bit slower):

NOT(NOT(A AND B AND C) AND (NOT (A AND B AND D)))
->
NOT((NOT(ABC)) AND (NOT (ABD)))
->
1 - ((NOT(ABC))(NOT(ABD)))
->
1 - (1 - ABC)(1 - ABC)
-> expanding the brackets
1 - (1 * 1 - ABC * 1 - ABD * 1 + ABC * ABD)
->
1 - (1 - ABC - ABD + ABCABD)
-> multiple out the bracket
1 - 1 + ABD + ABC - ABCABD

glad I could help ^^

Upvotes: 1

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