Is Only If Logic Misinterpreted in Mathematical Statements?

[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

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

 

[adjacent]

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.

 

[Atomised]

Thank you

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

 

[PeroK]

Thank you

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

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".

 

[Atomised]

I am sure PeroK and adjacent are both quite right but I can only conclude that there is misleading information out there e.g. http://en.wikibooks.org/wiki/Mathematical_Proof/Introduction/Logical_Reasoning seems to categorically state that 'P only if Q' equates to 'If Q Then P' What am I missing?

 

[PeroK]

I am sure PeroK and adjacent are both quite right but I can only conclude that there is misleading information out there e.g. http://en.wikibooks.org/wiki/Mathematical_Proof/Introduction/Logical_Reasoning seems to categorically state that 'P only if Q' equates to 'If Q Then P' What am I missing?

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"

 

[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

To say that "P is sufficient for Q" means "P cannot exist without Q" or "if P then Q"

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

 

[1MileCrash]

Thank you

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

"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.

 

[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.

 

[adjacent]

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

So true