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 double-headed 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."

Remember that a conditional statement has a one-way arrow () and a biconditional statement has a two-way arrow (). 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 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."

Example 5:  Rewrite each of the following sentences using "iff" instead of "if and only if."

Example 6:

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 double-headed arrow . The biconditional pq represents "p if and only if q," where p is a hypothesis and q is a conclusion.

Exercises

1.	Given: a: y - 6 = 9 b: y = 15 Problem: The biconditional ab represents which of the following sentences?
	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:

2.	Given: r: 11 is prime. s: 11 is odd. Problem: The biconditional rs represents which of the following sentences?
	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:

3.	Given: xy yx Problem: If both of these statements are true then which of the following must also true?
	(xy)(yx) xy x iff y All of the above. RESULTS BOX:

4.	Given: mn is biconditional Problem: Which of the following is a true statement?
	m is the hypothesis m is the conclusion n is a conditional statement n is a biconditional statement RESULTS BOX:

5.	Which of the following statements is biconditional?
	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:

Lessons on Symbolic Logic Negation Conjunction Disjunction Conditional Compound Biconditional Tautologies Equivalence Practice Exercises Challenge Exercises Solutions Related Activities Interactive Puzzle Printable Worksheets