Biconditional
IF AND ONLY IF

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In logic, a biconditional is a compound statement formed by combining two conditionals under "and."  Biconditionals are true when both statements (facts) have the exact same truth value. 

A biconditional is read as "[some fact]  if and only if [another fact]" and is true when
the truth values of both facts are exactly the same
 -- BOTH TRUE or BOTH FALSE.
  

Biconditionals are often
 used to form definitions.

Definition:  A triangle is isosceles if and only if the triangle has two congruent (equal) sides.

The "if and only if" portion of the definition tells you that the statement is true when either sentence (or fact) is the hypothesis.  This means that both of the statements below are true:

If a triangle is isosceles, then the triangle has two
 congruent (equal) sides. 
(true)

If a triangle has two congruent (equal) sides, 
then the triangle is isosceles.
  (true)

 

Mathematicians often use symbols and tables to represent concepts in logic.  The use of these variables, symbols and tables creates a shorthand method for discussing logical sentences.

Truth table biconditional (if and only if):
(notice the symbol used for "if and only if" in the table below)

T T T
T F F
F T F
F F T

A truth table is a pictorial representation of all of the possible outcomes of the truth value of a compound sentence.  Letters such as and are used to represent the facts (or sentences) within the compound sentence.

 I

 

REMEMBER:

IF AND ONLY IF is TRUE  
when both facts are
T  
or both facts are
F..