Biconditional Statements

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: (p
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al.gif q)
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/and.gif
(q
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al.gif p)

The compound statement (p
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al.gif q)
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/and.gif
(q
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al.gif p) 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

p
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al.gif q

q
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al.gif p

(p
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al.gif q)
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/and.gif
(q
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al.gif p)

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 (p
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al.gif q)
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/and.gif
(q
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al.gif p) 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 double-headed arrow
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal.gif . The biconditional p
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal.gif 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
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal.gif q.

p

q

p
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal.gif q

T

T

T

T

F

F

F

T

F

F

F

T

In the truth table above, p
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal.gif q 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 p
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal.gif q represent?

Solution:

The statement p
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal.gif q 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 one-way arrow (
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al.gif ) and a biconditional statement has a two-way arrow (
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal.gif ). We can use an image of a one-way street to help us remember the symbolic form of a
conditional statement, and an image of a two-way 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 a
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal.gif b as a sentence. Then determine its truth values a
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal.gif b.

Solution: The biconditional a
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal.gif b represents the sentence: "x + 2 = 7 if and only if x = 5." When x = 5, both a and b are
true. When x
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/not_equal
.gif 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
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal.gif b are listed in the table below.

a

b

a
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal.gif 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
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal.gif y as a sentence.

Solution: x
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal.gif y 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 r
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal.gif s as a sentence.

Solution:  r
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal.gif s 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 p
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al.gif q represent "If x + 7 = 11, then x = 5."
Let q
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al.gif p represent "If x = 5, then x + 7 = 11."

The statement p
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al.gif q is false by the definition of a conditional. The statement q
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al.gif p 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 r
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al.gif s is true by definition of a conditional. The statement s
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al.gif r 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 double-headed arrow
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal.gif . The biconditional p
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal.gif q 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 a
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal_transp.gif b represents which of the following sentences?


                                           Začátek formuláře

  (_) If y - 6 = 9, then y = 15.
  (_) y - 6 = 9 if and only if y = 15.
  (_) If y = 15, then y - 6 = 9.
  (_) None of the above.

  RESULTS BOX:
  _____________________________________________

                                            Konec formuláře


2.

Given:

r: 11 is prime.

s: 11 is odd.

Problem:

The biconditional r
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal_transp.gif s represents which of the following sentences?


                                           Začátek formuláře

  (_) If 11 is prime, then 11 is odd.
  (_) If 11 is odd, then 11 is prime.
  (_) 11 is prime iff 11 is odd.
  (_) None of the above.

  RESULTS BOX:
  _____________________________________________

                                            Konec formuláře


3.

Given:

x
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al_transp.gif y

y
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al_transp.gif x

Problem:

If both of these statements are true then which of the following must also true?


                                         Začátek formuláře

(_) (x
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al_transp.gif y)
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/and.gif
(y
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/condition
al_transp.gif x)
(_) x
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal_transp.gif y
(_) x iff y
(_) All of the above.

RESULTS BOX:
_____________________________________________

                                          Konec formuláře


4.

Given:

m
https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/biconditi
onal_transp.gif n is biconditional

Problem:

Which of the following is a true statement?


                                         Začátek formuláře

(_) m is the hypothesis
(_) m is the conclusion
(_) n is a conditional statement
(_) n is a biconditional statement

RESULTS BOX:
_____________________________________________

                                          Konec formuláře


5.

  Which of the following statements is biconditional?


                                           Začátek formuláře

  (_) I am sleeping if and only if I am snoring.
  (_) Mary will eat pudding today if and only if it is custard.
  (_) It is raining if and only if it is cloudy.
  (_) None of the above.

  RESULTS BOX:
  _____________________________________________

                                            Konec formuláře