
Conditional Statements 

Unit
9 > Lesson 4 of 11 
Example 1: 

Given:

p: I do my homework.

q: I get my allowance. 
Problem: 
What does pq represent? 

Solution: 
In Example 1, p represents, "I do my homework," and q represents
"I get my allowance." The statement pq
is a conditional statement which represents "If p, then q." 
Definition: 
A conditional statement, symbolized by pq,
is an ifthen statement in which p is a
hypothesis and q is a
conclusion.
The logical connector in a
conditional statement is denoted by the symbol .
The conditional is defined to be true unless a true hypothesis
leads to a false conclusion. A truth table for pq
is shown below. 


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

In the truth table above, pq
is only false when the hypothesis (p) is true and the conclusion (q) is false;
otherwise it is true. Note that a conditional is a compound
statement. Now that we have defined a conditional, we can apply it to Example 1.

Example 1: 

Given:

p: I do my homework.

q: I get my allowance. 
Problem: 
What does pq
represent? 

Solution: 
In Example 1, the sentence, "I do my homework" is the hypothesis
and the sentence, "I get my allowance" is the conclusion. Thus,
the conditional pq
represents the hypothetical proposition, "If I do my homework, then I get an
allowance." However, as you can see from the truth table above,
doing your homework does not guarantee that you will get an allowance! In
other words, there is not always a causeandeffect relationship between the hypothesis and conclusion of a conditional statement.

Example 2: 
Given:

a: The sun is made of gas.

b: 3 is a prime number. 
Problem: 
Write ab
as a sentence. Then construct a truth table for this conditional. 

Solution: The conditional ab
represents "If the sun is made of gas, then 3 is a prime number." 

a 
b 
ab 
T 
T 
T 
T 
F 
F 
F 
T 
T 
F 
F 
T 

In Example 2, "The sun is made of gas" is
the hypothesis and "3 is a prime number" is the conclusion. Note that the logical meaning of this conditional statement is not the
same as its intuitive meaning. In logic, the conditional is defined to be true unless a true hypothesis
leads to a false conclusion. The implication of ab is that: since
the sun is made of gas, this makes 3 a prime number. However, intuitively,
we know that this is false because the sun and the number three have nothing to do with one
another! Therefore, the logical conditional allows implications to
be true even when the hypothesis and the conclusion have no logical
connection. 
Example 3: 
Given: 
x: Gisele has a math assignment. 
y: David owns a car. 
Problem: 
Write xy
as a sentence. 

Solution: The conditional xy
represents, "If Gisele has a math assignment, then David owns a car.. 
In the following examples, we are given the truth values of the
hypothesis and the conclusion and asked to determine the truth value of the conditional.

Example 4: 
Given:

r: 8 is an odd number.

false

s: 9 is composite. 
true 
Problem: 
What is the truth value of rs? 

Solution: 
Since hypothesis r is false and conclusion s is true, the conditional rs is true. 
Example 5: 
Given:

r: 8 is an odd number.

false

s: 9 is composite. 
true 
Problem: 
What is the truth value of sr? 

Solution: 
Since hypothesis s is true and conclusion r is false, the conditional sr is false. 
Example 6: 

Given: 
p: 7^{2} = 49. 
true 
q: A rectangle does not have 4 sides. 
false 
r: Harrison Ford is an American actor. 
true 

s: A square is not a quadrilateral. 
false 
Problem: 
Write each conditional below as a sentence. Then indicate its truth value. 



1.  pq  If
7^{2} is equal to 49, then a rectangle does not have 4 sides.  false 
2.  qr  If
a rectangle does not have 4 sides, then Harrison Ford is an American actor.  true 
3.  pr  If 7^{2}
is equal to 49, then Harrison Ford is an American actor.  true 
4.  qs  If
a rectangle does not have 4 sides, then a square is not a quadrilateral.
 true 
5.  r~p  If Harrison Ford is
an American actor, then 7^{2} is not equal to 49.
 false 
6.  ~rp  If Harrison Ford is
not an American actor, then 7^{2} is equal to 49.
 true 

Note that in item 5, the conclusion is the negation of p. Also, in item 6, the hypothesis is the negation of r.
Summary:

A conditional statement, symbolized by pq, is an ifthen statement in which p is a
hypothesis and q is a conclusion. The conditional is defined to be true unless a true hypothesis
leads to a false 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.  Which of the following
is a conditional statement? 


2. 
Given:

r: You give me twenty dollars.

s: I will be your best friend. 
Problem:

Which of the following statements represents, "If you give me
twenty dollars, then I will be your best friend"?




3.  What is the truth
value of rs
when the hypothesis is false and the conclusion is true in Example 2? 


4. 
Given:

a: x is prime.

b: x is odd. 
Problem:

What is the truth value of ab
when x = 2?




5. 
What is the truth value of ab
when x = 9 in Exercise 4?



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