# Math: Explain IFF?
**URL:** https://boards.straightdope.com/t/math-explain-iff/445702
**Category:** Factual Questions
**Created:** [April 17, 2008, 7:25am UTC](https://boards.straightdope.com/t/math-explain-iff/445702 "2008-04-17T07:25:58Z")
**Posts on this page:** 8
**Page:** 1
### Author:  [@Jinx](https://boards.straightdope.com/u/Jinx)
#### Post date: [April 17, 2008, 7:25am UTC](https://boards.straightdope.com/t/math-explain-iff/445702/1 "2008-04-17T07:25:58Z")
IFF? If and only if. Like, “if” wasn’t good enough? WTF? :mad: It’s a Geometry thing; yes, I won’t understand! Understand I won’t. Not understand iff not I. Yeah…
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### Author:  [@Manduck](https://boards.straightdope.com/u/Manduck)
#### Post date: [April 17, 2008, 7:34am UTC](https://boards.straightdope.com/t/math-explain-iff/445702/2 "2008-04-17T07:34:26Z")
Example:
_I will go to the beach tomorrow if it’s sunny._
That leaves the possibility that I might go to the beach anyway, even if it’s cloudy.
_I will go to the beach tomorrow only if it’s sunny._
Leaves open the possibility that I might stay home, even if it’s sunny.
_I will go the the beach tomorrow if, and only if, it’s sunny_
Means that I will go to the beach if it’s sunny, and I will NOT go to the beach if it’s NOT sunny.
“A iff B” is a shorter way of saying “A if B and NOT A if NOT B”.
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### Author:  [@pulykamell](https://boards.straightdope.com/u/pulykamell)
#### Post date: [April 17, 2008, 7:48am UTC](https://boards.straightdope.com/t/math-explain-iff/445702/3 "2008-04-17T07:48:05Z")
IFF is what’s known as a biconditional. In other words, if P then Q, if Q then P. If not P then not Q, if not Q then not P.
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### Author:  [@The\_Them](https://boards.straightdope.com/u/The_Them)
#### Post date: [April 17, 2008, 10:53am UTC](https://boards.straightdope.com/t/math-explain-iff/445702/4 "2008-04-17T10:53:57Z")
You didn’t say what math symbology you’re using. In my world:
P⇔Q
is an abbreviation for
[(P⇒Q)∧(Q⇒P)]
This is great! I get to brag! 😉
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### Author:  [@MissMossie](https://boards.straightdope.com/u/MissMossie)
#### Post date: [April 17, 2008, 12:10pm UTC](https://boards.straightdope.com/t/math-explain-iff/445702/5 "2008-04-17T12:10:08Z")
Also, it’s not just a geometry thing. I’ve used it in every math (or logic) class that required proof writing.
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### Author:  [@Jurph](https://boards.straightdope.com/u/Jurph)
#### Post date: [April 17, 2008, 1:05pm UTC](https://boards.straightdope.com/t/math-explain-iff/445702/6 "2008-04-17T13:05:14Z")
It’s also used in engineering requirements documents (typically in state diagrams) when specifying that one function will occur if-and-only-if some other function occurs.
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### Author:  [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)
#### Post date: [April 17, 2008, 4:24pm UTC](https://boards.straightdope.com/t/math-explain-iff/445702/7 "2008-04-17T16:24:59Z")
There a big difference between “if” and “if and only if” (often abbreviated “iff”) (a difference which beginning math/logic students sometimes fail to appreciate, to their peril).
Compare: “You’ll die if you drink the cyanide.”
vs.
“You’ll die if and only if you drink the cyanide.”
With the second sentence, you have a reasonable shot at immortality.
P iff Q (also written with an arrow pointing both ways between the P and the Q) means that P and Q always go together: you can’t have one without the other.
if P, then Q (also written with a one-way arrow from P to Q, and which could also be worded as “Q if P,” or as “Q is true whenever P is true” or as “P implies Q”) means you’ll never see P without Q, but it doesn’t rule out the possibility that you’ll encounter Q without P.
One more more mathematical example:
A whole number is divisible by 2 if it ends in 0.
A whole number is divisible by 10 if and only if it ends in 0.
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### Author:  [@Frylock](https://boards.straightdope.com/u/Frylock)
#### Post date: [April 17, 2008, 4:31pm UTC](https://boards.straightdope.com/t/math-explain-iff/445702/8 "2008-04-17T16:31:24Z")
[QUOTE=Jinx]
IFF? If and only if. Like, “if” wasn’t good enough? WTF? :mad: It’s a Geometry thing; yes, I won’t understand! Understand I won’t. Not understand iff not I. Yeah…
[/QUOTE]
A iff B
means the same thing as
If A then B, and if B then A.
Tell me if that does not help.
-FrL-