contrapositive of A or B but not both... Clarification plz

hihihi9993

I understood "A or B but not both" as one of the followings:
1. A(O) B(X)
2. A(X) B(O)
3. A(X) B(X)

Then what is the contrapositive of it?
A/B = CONTRAPOSITIVE => /A + /B ?????
But my last statement (3) indicates /A and /B could fall under the definition of "A or B but not both".

Can someone clarify my thoughts please

GrecoRoman

A or B translates to /A -----> B, or is negate sufficient. The not both just makes it a biconditional /A <-----> B. So that would just be two worlds, A in, B out and B in, A out.

hihihi9993

Thank you!

hihihi9993

Actually! FYI for anyone who came into see this question... you can't apply both rules! Use only one of the rules to apply and the remaining variable simply becomes negated!
.
.
Therefore, A or B but not both
= /A ---> /B

akistotle

@d931n027h said:
Actually! FYI for anyone who came into see this question... you can't apply both rules! Use only one of the rules to apply and the remaining variable simply becomes negated!
.
.
Therefore, A or B but not both
= /A ---> /B

As @"work all week" wrote, "A or B but not both" = /A <----> B, A <----> /B.

The negation of the conditional statement is A and B.

batniki1

Good thing I saw this. Never thought that it's so confusing. Thanks for the summed up explanation @akistotle

hihihi9993

EDITED

hihihi9993

Oh wait this is always apart biconditional nvm! @akistotle thanks!

Sami

Hey so I think its a lot easier to understand this intuitively:

The first part of the sentence is:
A or B
-This means one of A or B has to be selected.
The possibilities for selection are:
1. A selected and B not selected
2. B selected and A not selected
3. A and B both selected.

The second part of the sentence is but not both. This adds an exception to our statement above.
The but not both is telling us either A or B has to be selected but there is a caveat to that - you cannot select both. This simply takes out our possibility number 3 (3) of having both A and B selected together. Now we only have two possible selections left.
1. A selected, B not selected (one world)
This means that if we know either A is selected or B not selected we know what the other has to be.
2. B selected, A not selected
This means if we know either B is selected or A not selected we are in this world.

Check this video out:
https://7sage.com/lesson/or-but-not-both/

When JY does it, he breaks it down the two sentences and then combines them.
A or B: /A---->B; contrapositive /B---->A
not both of A or B : A---->/B; contrapositive B---->/A

if you combine them it neatly lines up to:
A<----->/B and B<----->/A

vanessa fisher

@Sami
That's a great summary thanks.
Also, the negation of the entire bi-conditional would be A and B. Is that right?

Sami

@"vanessa fisher"

The negation is binary. So there are two negation of this statement.
A and B are both in like @akistotle said.
Also both A and B are both out.
Both of these possibilities are excluded from our "or but not both" rule.

hihihi9993

Thank you @Sami !