Iff is a shorthand for “if and only if”. Its logical symbol is \(\leftrightarrow\).
“A iff B” (\(A \leftrightarrow B\)) is quite distinct from “if A then B” (\(A \rightarrow B\)). Consider the stipulation “If the dog barks, then it will soon bite”. This would not obligate the dog to bark a warning before biting. The “if” relation isn’t symmetrical. As such, the dog might sometimes bite spontaneously, with no barking at all.
If we wanted to ensure that biting is always forewarned by barking, we would instead stipulate “Iff dog barks, then it will soon bite”. This is equivalent to
“the dog barks if and only if it will soon bite”
“If the dog barks then it will soon bite, and if the dog bites it will have barked beforehand”.
“The dog barks only when it will soon bite”
With “iff”, the implication runs in both directions.
Mathematics is the study of numbers and other ideal objects that can be described by axioms.