Disjunction

Circuits

Example 1: 

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

Baseball and Football

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: 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."

Harrison Ford

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."

Disjunction

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?
 
Amy played soccer or Bill played hockey.
Amy played soccer and Bill played hockey.
Amy did not play soccer and Bill played hockey.
None of the above.

RESULTS BOX:
 

2.   Which of the following statements is a disjunction?
 
~xy
xy
xy
None of the above.

RESULTS BOX:
 

3.   A disjunction is used with which connector?
 
And
Or
Not
None of the above.

RESULTS BOX:
 

4. If a is false and b is true, what is the truth value of a~b?
 
True
False
Not enough information was given
None of the above.

RESULTS BOX:
    

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?
 
r~s
r~s
rs
All of the above.

RESULTS BOX: