Formal negation of compound conditional

[Algorithm]

In the LG strategy guide, pg. 315 , I’m having trouble using the formal approach to negate #5.

Conditional:
If Sam goes , either Ruth or Tim, but not both, will go.

I can logically reason what the contrapositive should be, but when I use the block negation technique it doesn’t seem to work.

Conditional:
S —> (-R + T) or (-T + R)

If you treat the necessary condition from above as a block and negate the elements within, it doesn’t seem to work.


So The Contrapositive is:
(R + T) or (-R + -T) —-> -S
but I can’t figure out how to get here using the formal approach.


Thank you

[ohthatpatrick]

In general, what do we do with "or" when we're writing a contrapositive?


GIVEN:
A --> B or C

CONTRAPOS:
~B and ~C --> ~A


So,
GIVEN:
S —> (-R + T) or (-T + R)

CONTRAPOS:
~(~R + T) and ~(~T + R) --> ~S

You can 'distribute' a negative, kinda like in algebra when you're multiplying by -1.

[R or ~T] and [T or ~R] --> ~S

God that looks insane. I'm only answering in this style to humor you.

I was convinced that this is saying something wrong, but it actually is still getting at what we're trying to say, which is
IF they're both in or both out, then ~S

You could never have ~T and T. That's a contradiction.
You could never have R and ~R.

So the only way you could trigger that rule
[R or ~T] and [T or ~R] --> ~S
is if you had "R and T" or "~T and ~R".

Yikes.

Anyway, I deal with these differently.

When we're doing Grouping games, the most common rules are Friends (these two things must be together) and Enemies (these two things can't be together).

When two people have to be together, you put them in a circle:
(RT)

When two people can't be together, you put them in a circle and cross it out:
~(RT)

So I just use that convention for these rules:
S --> ~(RT)
(RT) --> S

If I have S, then R and T can't be together, forcing one IN and one OUT.
If RT are together (both in or both out), then I can't have S.

Hope this helps.

[Algorithm]

Your explanation is great! You cleared up my confusion with the "formal" negation , but I'll be using your practical approach going forward.

Thank you so much