Math: Explain IFF?

Factual Questions
1

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…

2

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”.

3

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.

4

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! :wink:

5

Also, it’s not just a geometry thing. I’ve used it in every math (or logic) class that required proof writing.

6

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.

7

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.

8

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-