I know that P <=> Q becomes (p => q) ^ (q => p)using double arrow elimination but how about the following ?

~(P <=> Q) becoming ~(p => q) ^ ~(q => p) ?


(~P <=> Q) becoming (~p => q) ^ (q => ~p)?

I want to make sure those are valid steps !


Your first example is incorrect because the negation distributes across the conjunction by way of De Morgan's Law. Therefore, the "AND" symbol needs to become an "OR," but otherwise, it is correct.

Your second example is correct as is.

  • Is there another rule to use for example 1 instead of DeMorgan's law? I don't want to get an "OR", its kind of hard to work with them!
    – Questions
    Dec 14 '14 at 22:07
  • @Questions Nothing comes to mind. Obviously, you could write it as a "NAND" statement, but that's the only thing I can think of and it doesn't seem that useful.
    – Geoffrey
    Dec 14 '14 at 22:11
  • @Questions Actually, "NOT(P IFF Q)" is logically equivalent to "P XOR Q," but again, I doubt you'll find that useful.
    – Geoffrey
    Dec 14 '14 at 22:20
  • I am trying to use these in derivations but XOR can not be symbolized, it has to be from the rules.
    – Questions
    Dec 14 '14 at 22:26
  • I marked your answer since you did answer my question, however if anyone can answer my second question on the comments above, I can at least give +1 for appreciation.
    – Questions
    Dec 15 '14 at 3:56

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