# If only I understood only if

1. Apr 5, 2014

### Atomised

1. Question in book is:

State which of the following are true / false

a) n = 3 only if n^2 - 2n - 3 = 0

b) n^2 - 2n - 3 = 0 only if n=3

c) If n^2 - 2n - 3 = 0 then n = 3

The solutions it gives are

a) True b) False c) False

2. My assumptions

P only if Q is logically equivalent to If Q then P

3. The attempt at a solution

Taking P to be n=3
And Q to be n^2 - 2n - 3 = 0

Restating the question

a) Q implies P

b) P implies Q

c) Q implies P

Since a) & c) are logically equivalent they must have the same answer yet the printed solution states otherwise.

What am I not getting?

Many thanks

2. Apr 5, 2014

a) If n=3 then n^2 - 2n - 3 = 0 - Yes if n is equal to 3 , then n^2- 2n - 3 is equal to zero
b)If n^2 - 2n - 3 = 0 then n=3 - No. n can be -1 too.
c)If n^2 - 2n - 3 = 0 then n = 3 - This means the same as b.

3. Apr 5, 2014

### Atomised

Thank you

So X only if Y is logically equivalent to if X then Y?

4. Apr 5, 2014

### PeroK

Suppose: X only if Y:

X True, Y True (Yes)
X True, Y False (No)
X False, Y True (Yes)
X False, Y False (Yes)

Suppose: If X, then Y:

You can confirm that the above holds. So, yes they are the same.

Note: I've used "yes" for this combination does not break the rule; and, "no" for breaks the rule.

Note "only if" is really only used to test your logical thinking. Because of the above equivalence, in practice most people use "if X then Y".

You can also check from the above table that these are also equivalent to "If not Y, then not X".

5. Apr 5, 2014

6. Apr 5, 2014

### PeroK

It's a bit misleading because it doesn't say clearly what the "if" and "only if" apply to. The way to interpret that truth table is:

P iff Q means "P if Q" and "P only if Q"; which is equivalent to "Q => P" and "P => Q"

7. Apr 5, 2014

### AlephZero

http://en.wikibooks.org/wiki/Mathematical_Proof/Introduction/Logical_Reasoning gives the wrong symbols under the heading "Implication types", and later it uses talks about "existence" instead of "truth". Statements like
are at best very confusing IMO.

I would treat it the same as the rest of Wikipedia, i.e. assume what is says is true only if you already know it is true

For a better explanation, see http://www.math.csusb.edu/notes/logic/lognot/node1.html and http://www.math.csusb.edu/notes/logic/eequiv/eequiv.html

8. Apr 5, 2014

### 1MileCrash

"X if Y" means "if Y then X," clearly.

I'll also assume that you know that "X if and only if Y" means "if X then Y, and if Y then X".

Therefore we can conclude that "X only if Y" must be "if X then Y" because adding this to "X if Y" adds that implication to its logical meaning.

9. Apr 5, 2014

### Atomised

AlephZero - thanks for busting wikipedia - I should know better than to be misled by it.

1MileCrash - I am now having the aha moment... of course the 'only if' is the other direction from 'if' in iff, also a brilliant way of remembering it thank you, job done.

10. Apr 6, 2014

:rofl: So true