Simple question that keeps aggravating me

[greatwhiteshark100]

Hey guys, you've been a great help so far.

I got this little LG thing that always troubles me.

If A and B, then C.

Does that mean
A->C
B->C
or
A&B->C

If C, then A and B.
Does that mean
C->A&B
or
C->A
C->B

Thanks！

[aileenann]

No worries - happy to help.

If A and B, then C translates to: A & B-> C

Think about it for a second - if you only know A but don't know about B, that's not enough to conclude C. That's why we need the A & B condition.

Your second expression, If C, then A and B. is a little more complicated It does mean C >- A & B, but here we can also express that as C->A and C->B so long as we keep both of the statements. This is because here C is enough to know that A happens and that B happens, so we can express that as individual relationships between C and A and C and B, or as a relationship between C and (A & B). They are functionally, and logically, equivalent.

Does that mean
C->A&B
or
C->A
C->B

[thepowell]

I've been struggling with the same, and particularly when paired with "only if":

Garry gets into the pool only if Chris and Even get into the pool.

To me, that's:
If (C+E) then G
C+E>G
Not G > Not C+E

However, my book says:
G > C
G > E

But it doesn't say "only if Chris OR Evan" so why is it split? (And why is the order reversed?)

[timmydoeslsat]

I've been struggling with the same, and particularly when paired with "only if":

Garry gets into the pool only if Chris and Even get into the pool.

To me, that's:
If (C+E) then G
C+E>G
Not G > Not C+E

However, my book says:
G > C
G > E

But it doesn't say "only if Chris OR Evan" so why is it split? (And why is the order reversed?)

You win the lottery only if you play.

Win Lottery ---> Play

Only if = necessary condition.

So, your sentence:

Garry gets into the pool only if Chris and Even get into the pool.

Garry ---> Chris and Evan

The contrapositive is simply switch and negate.

~ Chris or ~ Evan ---> ~ Garry


Think of it at a more fundamental level. We know that Garry being in the pool requires Chris and Evan.

Do we know what happens if Chris is not in the pool?

Garry is not in!

Do we know what happens if Evan is not in the pool?

Garry is not in!

Do we know what happens if Chris is not in the pool and Evan is not in the pool?

Garry is not in!

That is what the contrapositive is telling us.

~Chris or ~Evan ---> ~Garry

The absence of just one of Chris and Evan is sufficient to knowing that Garry is not in the pool.

[ManhattanPrepLSAT1]

Nice explanations Timmy!

Let me just add one more point about statements involving "and" and "or."

If the statement read, "Gary gets in the pool only if Chris or Evan gets in the pool," we could notate it:

G --> C or E

This statement we could not break into separate statements.

As is, "Gary gets in the pool only if Chris and Evan get in the pool," we can notate it:

G --> C + E

This statement we can break up:

G --> C
G --> E

In summary...

"and" in the trigger:

A + B ---> C

we cannot break into separate statements.

"and" in the outcome:

A --> B + C

we can break into separate statements

A --> B
A --> C

"or" in the tigger:

A or B --> C

we can break into separate statements

A --> C
B --> C

"or" in the outcome:

A --> B or C

We cannot break into separate statements.