
Biconditional Statements 

Unit
9 > Lesson 6 of 11 
Example 1: Examine the sentences below. 
Given: 
p: A polygon is a triangle. 
q: A polygon has exactly 3 sides. 
Problem: 
Determine the truth values of this statement:
(pq)(qp) 


The
compound statement
(pq)(qp)
is a conjunction of two conditional
statements. In the first conditional, p is the hypothesis and q is the
conclusion; in the second conditional, q is the hypothesis and p is the
conclusion. Let's look at a truth table for this
compound statement.

p 
q 
pq 
qp 
(pq)(qp) 
T 
T 
T 
T

T

T 
F 
F 
T

F

F 
T 
T 
F

F

F 
F 
T 
T

T

In the truth table above, when p
and q have the same truth values, the compound statement
(pq)(qp)
is true. When we combine two conditional statements this way, we have a 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 doubleheaded arrow . The
biconditional pq 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
pq. 


p 
q 
pq 
T 
T 
T 
T 
F 
F 
F 
T 
F 
F 
F 
T 

In the truth table above, pq
is true when p and q have the same
truth values, (i.e., when either both are true or both are false.) Now that the biconditional has been defined, we can look at a modified version
of Example 1.

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


Note that in the biconditional above, the hypothesis is: "A polygon is a triangle"
and the conclusion is: "It has exactly 3
sides." It is helpful to think of the biconditional as a conditional
statement that is true in both directions.



Remember that a conditional statement has a oneway
arrow ()
and a biconditional statement has a twoway arrow ().
We can use an image of a oneway street to help us remember the
symbolic form of a conditional statement, and an image of a twoway
street to help us remember the symbolic form of a biconditional
statement. 



Let's look at more examples of the biconditional.
Example 2:
Given: 
a: x + 2 = 7 
b: x = 5 
Problem: 
Write ab
as a sentence. Then determine its truth values ab. 
Solution:
The biconditonal ab
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 ab
are listed in the table below.

a 
b 
ab 
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 xy
as a sentence. 
Solution: xy
represents the sentence, "I am breathing if and only if I am alive."
Example 4:
Given: 
r: You passed the exam. 
s: You scored 65% or higher. 
Problem: 
Write rs
as a sentence. 
Solution: rs
represents, "You passed the exam if and only if you scored 65% or
higher."

Mathematicians abbreviate "if and only if" with "iff." In Example 5, we will rewrite each sentence from
Examples 1 through 4 using this abbreviation.

Example 5: Rewrite each of the following sentences using "iff" instead
of "if and only if."
if and only if 
iff 
A polygon is a triangle if and only if it has exactly 3 sides. 
A polygon is a triangle iff it has exactly 3 sides. 
I am breathing if and only if I am alive. 
I am breathing iff I am alive. 
x + 2 = 7 if and only if x = 5. 
x + 2 = 7 iff x = 5. 
You passed the exam if and only if you scored 65% or higher. 
You passed the exam iff you scored 65% or higher. 
When proving the statement p iff q, it is equivalent to proving both of the statements "if p, then q" and "if q, then
p." (In fact, this is exactly what we did in Example 1.) In each
of the following examples, we will determine whether or not
the given statement is biconditional using this method.

Example 6:
Given: 
p: x + 7 = 11 
q: x = 5 
Problem: 
Is this sentence biconditional? "x + 7 = 11 iff x = 5." 
Solution:
Let pq
represent "If x + 7 = 11, then x = 5."
Let qp
represent "If x = 5, then x + 7 = 11."
The statement pq
is false by the definition of a conditional. The statement qp
is also false by the same definition. Therefore, the sentence "x + 7 = 11 iff x = 5"
is not biconditional.

Example 7:
Given: 
r: A triangle is isosceles. 

s: A triangle has two congruent (equal) sides. 
Problem: 
Is this statement biconditional? "A triangle is isosceles if and only if
it has two congruent (equal) sides." 
Solution:
Yes. The statement rs
is true by definition of a conditional. The statement sr
is also true. Therefore, the sentence "A triangle is isosceles if and only if
it has two congruent (equal) sides" is biconditional.

Summary:

A biconditional statement is
defined to be true whenever both parts have the same truth value. The biconditional
operator is denoted by a doubleheaded arrow . The
biconditional pq
represents "p if and only if q," where p is a hypothesis and q
is a conclusion.

Exercises
Directions: Read each question below. Select your answer by clicking on its button. Feedback to your answer
is provided in the RESULTS BOX. If you make a mistake, choose a different button.

1. 
Given:

a: y  6 = 9

b: y = 15 
Problem:

The biconditional ab
represents which of the following sentences?




2. 
Given:

r: 11 is prime.

s: 11 is odd. 
Problem:

The biconditional rs
represents which of the following sentences?




3. 
Given: 
xy 
yx 
Problem: 
If both of these statements are true then which of the following must
also true? 



4. 
Given:

mn
is biconditional

Problem:

Which of the following is a true statement?




5.  Which of the following
statements is biconditional?



This lesson is by Gisele Glosser. You can find me on Google.
