what is the point of iff (if and only iff)?

Question

In several math classes I have taken, my instructors use iff to mean if and only if. Am I the only one who think this is retarded? I always thought "only if" was already implied by just writing "if", if it was something other than if , wouldn't I have written that.

Answer 1

"If and only if" is an extremely important logical qualifier.

A person is a US citizen if he or she was born in US territory.
A person is a US citizen if and only if he or she was born in US territory.

The first statement is true. The second statement is false, because a person is also a US citizen if he or she is born to at least one US citizen parent, or if she or she becomes naturalized.

"If and only if" is also important because logical operations on an iff statement (such as negation, converse, contrapositive) have different results than the same operation on a simple if statement.

Answer 2

Mathematics defines "if' differently than most people think of it in English.

The mathematical statement:

If A, then B

Is a true statement if A is true and B is false, which is expected.

However, the tricky part is that if A is false, the statement is ALSO considered to be true...no matter what B is.

So the statement:

If Bill Clinton is currently president of the United States, the sky is green.

Is considered to be a true statement, mathematically! Because the first part is false, it doesn't matter what the second part says.

Now, mathematically, "only if" is the OPPOSITE of if; that is:

A only if B.

Is true if B is true and A is true, or if B is false (regardless whether A is true and B is false).

So when you put together if and only if, you end up with a statement that is symmetrical:

If A, then B ... AND .... A only if B.

which is the same as:

A iff B

Unlike the other 2, this statement is true when A and B are both true or when A and B are both false. If A is true and B is false, or vice versa, the statement is false.

Example:

Take the statement:

"A triangle T with sides of length a, b, and c is a right triangle IF AND ONLY IF a² + b² = c²."

This statement can be be broken into two statements:

"IF a triangle T with sides of length a, b, and c is a right triangle, THEN a² + b² = c²."

and

"IF a² + b² = c², THEN T is a right triangle."

Both these statements are true, so the IFF statement is true.

However, consider the statement:

"A quadrilateral Q is a square IF AND ONLY IF Q is a parallelogram."

That becomes two statements:

"IF a quadrilateral Q is a square, THEN Q is a parallelogram."

That statement is true, because all squares are parallelograms.

But the reverse statement is false:

"IF a quadrilateral Q is a parallelogram, THEN Q is a square."

Since one of the statements is false, the original IF AND ONLY IF statement is also false.

Answer 3

if and only if means there is no other possible way of doing something.

You may drive and operate a vehicle if and only if you have a license. Basically, no other exception would let you drive unless it was against the law...or math laws.

if just means if but there are other ways to do stuff. iff means it's the ONLY way to do stuff

Answer 4

There is a difference, actually. If is true, then is true does not impy that, if is true, then is true. For example, "if Bill is an eagle, then Bill can fly" does not imply that "if Bill can fly, then Bill is an eagle".

However, by definition, iff is true, then is true DOES impy that, if is true, then is true. "If n = 2, then n is a prime even number" and "if n is a prime even number, then n = 2" are both true.

Answer 5

It's just convention!

x = y iff x > 0

x is equal to y if and only if x is greater than zero.

Or, as you said, x is equal to y if x is greater than zero.

Yours sounds correct; however, it's standard to use iff.

Answer 6

We say 'if P, then Q' when P implies Q. This is the same as saying 'Q, if P'.

We say 'P only if Q' when Q implies P. This is very different than P implying Q.

For example,

'If you are human, then you have one head' is true.
'If you have one head, then you are human' is false.

So
'You have one head if you are human' is true. But
'You have one head only if you are human' is false.