BICONDITIONAL

Definition:

A biconditional statement is defined to be true whenever both parts have the same truth value. The biconditional operator is denoted by a double-headed arrow ↔. The biconditional p↔q represents "p if and only if q," where p is a hypothesis and q is a conclusion. The following is a truth table for biconditional p↔q. In the truth table above, p↔q is true when p and q have the same truth values, (i.e., when either both are true or both are false.) p q p↔q T T T T F F F T F F F T

Example 1:
Given: p: A polygon is a triangle. q: A polygon has exactly 3 sides. Problem: What does the statement p↔q represent?
Solution:The statement p↔q represents the sentence, "A polygon is a triangle if and only if it has exactly 3 sides."

Example 2:

Given:	a: x + 2 = 7
	b: x = 5
Problem:	Write a↔b as a sentence. Then determine its truth values a↔b.

Solution: The biconditonal a↔b represents the sentence: "x + 2 = 7 if and only if x = 5." When x = 5, both a and b are true. When x ↔5, both a and b are false. A biconditional statement is defined to be true whenever both parts have the same truth value. Accordingly, the truth values of a↔b are listed in the table below.

a	b	a↔b
T	T	T
T	F	F
F	T	F
F	F	T

Example 3:

Given:	x: I am breathing
	y: I am alive
Problem:	Write x↔y as a sentence.

Solution:  x↔y represents the sentence, "I am breathing if and only if I am alive."