If and Only if & Only if Questions!

[Echx73]

Team PowerScore,

I was wondering if you could help me out on a few questions. I do not feel comfortable when writing out the the rules for these types of questions, along with the contrapositive. Would you be kind enough and show me how you would write each rule, along with writing out the contrapositive, please? Thank you so much for your help.

T is in group #1 if and only if R is in group #2

T is not in group #1 if and only if R is in group #2

T is in group #1 if and only if R is not in group #2

T is not in group #1 if and only if R is not in group #2


T is in group #1 only if R is in group #2

T is not in group #1 only if R is in group #2

T is in group #1 only if R is not in group #2

T is not in group #1 only if R is not in group #2

[Ladan Soleimani]

Hi Echx73

Your first set of rules are double arrows or biconditionals. The 'if and only if' mean that each term is a sufficient and a necessary condition because 'if' indicates a sufficient condition and 'only if' indicates a necessary condition. The way the rules are diagrammed is by combining the two conditionals into one with a double arrow.
For example, your first rule:

T is in group #1 if and only if R is in group #2

can be broken into two conditionals:

T is in group #1 if R is in group #2 diagrammed as R2 T1
and
T is in group #1 only if R is in group #2 diagrammed as T1 R2

Combined we write these as
T1 R2 Contrapositive: T1 R2
You negate both sides to get the contrapositive. Since the arrow goes both ways it is unnecessary to flip the contrapositive, but you can if you want: it will have the same meaning. The rest of this set would be similar.

T is not in group #1 if and only if R is in group #2
T1 R2 Contrapositive: T1 R2

T is in group #1 if and only if R is not in group #2
T1 R2 Contrapositive: T1 R2

T is not in group #1 if and only if R is not in group #2
T1 R2 Contrapositive: T1 R2

You second set are more straightforward conditionals. "only if" is a necessary indicator in all of these, so what it modifies will be on the right side of the arrow. Then you just flip and negate to get the contrapositive.

T is in group #1 only if R is in group #2
T1 R2 Contrapositive: R2 T1

T is not in group #1 only if R is in group #2
T1 R2 Contrapositive: R2 T1

T is in group #1 only if R is not in group #2
T1 R2 Contrapositive: R2 T1

T is not in group #1 only if R is not in group #2
T1 R2 Contrapositive: R2 T1

I hope this helps!
Ladan

[Echx73]

Ladan,

Thank you for the help above. I needed to double check my negations and orders. I do, though, have one more question on this subject.

I came along a game where the conditional statement went as follows: D can be displayed only if K is displayed.
I understand how to draw the rule, but what is throwing me off is the "Can." I have always seen the "only if" statements use 'Is.' Is there a difference between saying D can be displayed or D is displayed.... only if K is displayed? Maybe I reading too much into the statement.

Thank you!

[Lucas Moreau]

Hello, Echx,

The "can" changes this from a straightforward conditional statement, yes.

Since it's not D is displayed only if K is displayed, it's D can be displayed only if K is displayed, that can be rephrased to make it better. You can almost always rephrase conditional statements to make them easier. In this case, the statement should convert into:

"If D is displayed, K is displayed."

This works if you break it down. Assume there are two possibilities, D being displayed and D not being displayed. If we don't see D, we know nothing about K, but if we see D being displayed, we know that K also must be being displayed as well - that's the only way D can be displayed.

Hope that helps,
Lucas Moreau