Formal Logic Diagram involving either or HELP!

[karin.yoo]

I am having difficulty in making FL diagram involving either or. How do you make FL diagram and its contrapositive with the 3 cases listed below? Thanks in advance
**Correct any mistakes please!

1. Either A or B or both
Ex) if either A or B or both are present, then so is C.
My interpretation) A or B --> C
(since either or already consists of the "or both" element)
Contrapositive) ~C --> ~A and ~B

2. Either A or B but not both
Ex) if either A or B but not both are present, then C is.
My interpretation) A --> C, B--> C
Contrapositive) ~C --> ~A, ~C --> ~B

3. If it is not the case that both A and B are present, then C is.
My interpretation) A or B --> C, ~A and ~B --> C
Contrapositive) ~C --> ~A and ~B, ~C --> A or B


**I have previously posted my question on a forum, but everybody has given me different opinions and explanations, which worsened my confusion.
Are my interpretations and contrapositives above correct?
Please clarify.

Thank you in advance!

[maryadkins]

1. Either A or B or both
Ex) if either A or B or both are present, then so is C.
My interpretation) A or B --> C
(since either or already consists of the "or both" element)
Contrapositive) ~C --> ~A and ~B

Yes!

2. Either A or B but not both
Ex) if either A or B but not both are present, then C is.
My interpretation) A --> C, B--> C
Contrapositive) ~C --> ~A, ~C --> ~B

Close, but we need to get the "but not both" in there. So let's add that to what you have:

A + ~B --> C
B + ~A --> C

If both are present, we don't know anything, because our conditional limits its trigger to "but not both" being there.

3. If it is not the case that both A and B are present, then C is.
My interpretation) A or B --> C, ~A and ~B --> C
Contrapositive) ~C --> ~A and ~B, ~C --> A or B

I can't imagine you'd ever actually see this. But if you do, it's:

~A or ~B --> C
~C --> A and B

You don't need "A or B --> C" because that's only true if one of them is not in, which is already captured by the statement I just wrote above. Again, I doubt you'll see this .

Hope this helps!

[easyvoca]

karin.yoo wrote:
3. If it is not the case that both A and B are present, then C is.
My interpretation) A or B --> C, ~A and ~B --> C
Contrapositive) ~C --> ~A and ~B, ~C --> A or B

maryadkins replied :
I can't imagine you'd ever actually see this. But if you do, it's:

~A or ~B --> C
~C --> A and B

HERE'S MY QUESTION.

Super Prep test B
Section 2 Q7-12
rule 4 : If it is not the case that the park contains both laurels and oaks, then it contains firs and spruces.

Does this mean
-L or -O or -L and -O -> F AND S ?
If not, I can't crack this one.
Could anybody explained it to me?

Thanks in advance.

[timmydoeslsat]

rule 4 : If it is not the case that the park contains both laurels and oaks, then it contains firs and spruces.

Does this mean
-L or -O or -L and -O -> F AND S ?

You are technically correct in your diagram of that rule.

To make it easier to read during the game, I would diagram it as this:

~ [L and O] ---> [F and S]

In other words, this game requires at least one of those pairs in at all times.

If you do not have the LO pair, you must have the FS pair.

You could have both pairs in the game of course, but never could both pairs be out.