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Hey Guys. I just wanted to see if someone can tell me how to denote "A or B but not both" in formal logic with both sides of the contra-positive. Thanks in advance.

Edit: I meant to write If X -> then A or B but not both. ( sorry).

A +B ---> not x

X ----> A or B

jitsubruin wrote:A +B ---> not x

X ----> A or B

not (A or B) ----> not X

dingbat wrote:

jitsubruin wrote:A +B ---> not x

X ----> A or B

not (A or B) ----> not X

X --- > [(A or B) & ~(A & B)]

Cerebro wrote:

dingbat wrote:

jitsubruin wrote:A +B ---> not x

X ----> A or B

not (A or B) ----> not X

X --- > [(A or B) & ~(A & B)]

Dingbat's response = contrapositive to jitsu's second statement.

Cerebro's X -> ~(A & B) = contrapositive to jitsu's first statement.

Everyone agrees!

X -> (A or B) and (~A or ~B)

There are a few ways to write it (as seen above), but I find this to be the most straight-forward, the easiest to remember, and the easiest to work with.

Celebro is right, X ---> [ (A OR B) & ~(A & B) ] is the correct logical formulation.

I’m not sure I understood the suggestions for the contrapositive above. So I’ll just suggest my own.

X ---> [ (A OR B) & ~(A & B) ]

So reverse and negate both sides .
~ [ (A OR B) and ~(A & B) ] ---> ~X

Then put the negation inside the brackets, and throw away the brackets.
~(A OR B) OR ~( A & B) ----> ~X

Then put the negations inside the parentheses.
(~A & ~B ) OR (~A OR ~B) ---> X

so (~A & ~B ) OR (~A OR ~B) ---> X is the contrapositive.

TylerJonesMPLS wrote: so (~A & ~B ) OR (~A OR ~B) ---> X is the contrapositive.

I don't have time to see where you went wrong, but that's not the correct contrapositive. Especially since it's redundant - the first parenthetical is included in the second parenthetical. And you forgot the negation of X.

I'm going back to my formulation because there's no need to mess with moving negations around:
X -> (A OR B) AND (~A OR ~B) (which is the same as Celebro, if you move the negation inside the second parenthetical instead of keeping it outside - it's more symmetrical that way)
(A OR B) is saying at least one of A or B; (~A OR ~B) is the same as saying not both A and B. Put them together and you get at least one, but not both.
Contrapositive:
(A AND B) OR (~A AND ~B) -> ~X
If I have both, or if I have neither, I don't have X.

The question is how to translate the sentence "If X then A or B but not both" into propositional logic, and to show the contrapositive.
Celebro's answer to the question of how to put the sentence in propositional logic is obviously correct: X ---> [(A OR B) AND ~(A AND B)]
(One can use DeMorgan's laws to derive (~A OR ~B) from ~(A AND B), but that requires a second step in the logic.)
I apologize for the typos and miscounting the negations in my earlier post; I wrote it out too quickly and carelessly. I will will begin with Celebro's formulation, and derive the contrapositive in two ways, because one may be easier to see than the other.

1. X > [(A or B) and ~(A and B)]
2. ~ [(A or B) and ~(A and B)] > ~X
3. [~(A or B) or ~~(A and B)] > ~X
4. [~(A or B) or (A and B)] > ~X
5. [(~A and ~B) or (A and B)] > ~X

1. X > [(A or B) and ~(A and B)]
2. X > [~(~A and ~B) and ~(A and B)]
3. ~[~(~A and ~B) and ~(A and B)] > ~X
4. [~~(~A and ~B) or ~~(A and B)] > ~X
5. (~A and ~B) or (A and B)] > ~X