Learn About Disjunction With The Following Examples And Interactive Exercises.
Example 1:
| Given: | p: Ann is on the softball team. |
| q: Paul is on the football team. | |
| Problem: | What does p q 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
|
 |
|
|
|
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 | x y |
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. |
|
|||||
