
Disjunction 

Unit
9 > Lesson 3 of 11 
Example 1: 

Given: 
p: Ann is on the softball team. 
q: Paul is on the football team. 
Problem: 
What does pq
represent? 

Solution: 
In Example 1, statement p represents, "Ann is on the softball team" and statement q represents,
"Paul is on the football team." The symbol is a
logical connector which means "or." Thus, the compound statement
pq represents
the sentence, "Ann is on the softball team or Paul is on the football team." The statement
pq is a disjunction.

Definition: 
A disjunction is a compound statement formed by joining
two statements with the connector OR. The disjunction "p or q" is
symbolized by pq.
A disjunction is false if and only if both statements are false;
otherwise it is true. The truth values of pq
are listed in the truth table below.



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

Example 2: 


Given: 
a: A square is a quadrilateral. 
b: Harrison Ford is an American actor. 
Problem: 
Construct a truth table for the disjunction "a or b." 

Solution: 
a 
b 
ab 
T 
T 
T 
T 
F 
T 
F 
T 
T 
F 
F 
F 

Example 3: 
Given: 
r: x is divisible by 2. 
s: x is divisible by 3. 
Problem: 
What are the truth values of rs? 

Solution: 
Each statement given in this example represents an
open sentence,
so the truth value of rs
will depend on the replacement values of x as shown below. 

If x = 6, then r is true, and s is true. The disjunction
rs is true. 
If x = 8, then r is true, and s is false. The disjunction
rs is true. 
If x = 15, then r is false, and s is true. The disjunction
rs is true. 
If x = 11, then r is false, and s is false. The disjunction
rs is false. 
Example 4: 
Given: 
p: 12 is prime. 
false 
q: 17 is prime. 
true 
r: 19 is composite. 
false 
Problem: 
Write a sentence for each disjunction below. Then indicate if it is true or false. 


1. 
pq 
12 is prime or 17 is prime. 
true 
2. 
pr 
12 is prime or 19 is composite. 
false 
3. 
qr 
17 is prime or 19 is composite. 
true 

Example 5: 
Complete a truth table for each disjunction below. 

1. 
a or b 
2. 
a or not b 
3. 
not a or b 


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


a 
b 
~b 
a~b 
T 
T 
F 
T 
T 
F 
T 
T 
F 
T 
F 
F 
F 
F 
T 
T 


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


Students sometimes confuse conjunction and disjunction. Let's look at an example
in which we compare the truth values of both of these compound statements.

Example 6: 
Given: 
x: Jayne played tennis. 
y: Chris played softball. 
Problem: 
Construct a truth table for conjunction "x and y" and disjunction "x or y." 


Solution:
x 
y 
xy 
xy 
T 
T 
T 
T 
T 
F 
F 
T 
F 
T 
F 
T 
F 
F 
F 
F 
With a conjunction, both statements must be true for the conjunction to be
true; but with a disjunction, both statements must be false for the disjunction to be
false. One way to remember this is with the following mnemonic: 'And’
points up to the sand on top of the beach, while ‘or’ points down to the ore
deep in the ground.

Summary:

A disjunction is a compound statement formed by joining
two statements with the connector OR. The disjunction "p or q" is
symbolized by pq.
A disjunction is false if and only if
both statements are false; otherwise it is true.

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
sentences is a disjunction? 


2.  Which of the
following statements is a disjunction? 


3.  A disjunction is used with which
connector? 


4. 
If a is false and b is true, what is the truth value of a~b?



5. 
Given:

r: y is prime.

s: y is even. 
Problem:

Which of the following is a true statement when y is replaced by 3?




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