Logic: is ( A && !(B || C)) || ( B || C ) the same as ( A || B || C )?

Question

I've encountered some obj-c code and I'm wondering if there's a way to simplify it: #if ( A && !(B || C)) || ( B || C ) is this the same as? #if ( A || B || C ) If not, is there another way to formulate it that would be easier to read? [edit] I tried the truth table before asking the question, but thought I had to be missing something because I doubted that Foundation.framework/Foundation.h would employ this more complex form. Is there a good reason for it? Here's the original code (from Foundation.h): #if (TARGET_OS_MAC && !(TARGET_OS_EMBEDDED || TARGET_OS_IPHONE)) || (TARGET_OS_EMBEDDED || TARGET_OS_IPHONE)

Answer 1

YES!!!!

Here is a direct proof as well as a truth table. Enjoy!

(A and not ( B or C)) or (B or C) = <--- given

(A and not B and not C) or (B or C) = <--- DeMorgan's

Distributive

A or (B or C) and [not B or (B or C)] and [not C or (B or C)] =

Associative

A or (B or C) and [not B or B or C] and [not C or B or C] =

Cancellation/Asportion Property : X or not X must be true; third var gets eaten

A or (B or C) and True and True

Absoprtion Property: outcome depends entirely on A or (B or C)

A or (B or C)

Truth Table:

A B C B or C !(B or C) A and !(B or C) RHS A or B or C

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T T T T F F T T

T T F T F F T T

T F T T F F T T

T F F F T T T T

F T T T F F T T

F T F T F F T T

F F T T F F T T

F F F F T F F F