Learn About Disjunction With The Following Examples And Interactive Exercises.
|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.
|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."|
|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.
|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.
|Given:||x: Jayne played tennis.|
|y: Chris played softball.|
Construct a truth table for conjunction "x and y" and disjunction "x or y."
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.
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?|