One example of a biconditional statement is "a triangle is isosceles if and only if it has two equal sides." A biconditional statement is true when both facts are exactly the same, either both true or both false. Biconditional statements are created to form mathematical definitions.
A biconditional allows mathematicians to write two conditionals at the same time. The example, "a triangle is isosceles if and only if it has two equal sides," means that "if a triangle is isosceles, then it has two equal sides" and that "if a triangle has two sides, then it is isosceles."
The four possibilities of a biconditional statement can be represented in a truth table. Each fact in the statement is represented by a different letter. For example, if fact "a" is true and fact "b" is true, then the biconditional is true. If both "am" and "b" are false, then the biconditional is also true. However, if "a" is true and "b" is false, or if "a" is false and "b" is true, then the biconditional statement is false.
The phrase "if and only if" is sometimes abbreviated as "iff." This abbreviation was first used in print in the book "General Topology" by John L. Kelley in 1995. He credits Paul Halmos for the creations of "iff."