Disjunction Example 1: Given: p: Ann is on the softball team. q: Paul is on the football team. Problem: What does p https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif 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 https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif is a logical connector which means "or." Thus, the compound statement p https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif q represents the sentence, "Ann is on the softball team or Paul is on the football team." The statement p https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif q 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 p https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif q. A disjunction is false if and only if both statements are false; otherwise it is true. The truth values of p https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif q are listed in the truth table below. p q p https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif q 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 a https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif b 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 r https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif s? Solution: Each statement given in this example represents an open sentence, so the truth value of r https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif s will depend on the replacement values of x as shown below. If x = 6, then r is true, and s is true. The disjunction r https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif s is true. If x = 8, then r is true, and s is false. The disjunction r https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif s is true. If x = 15, then r is false, and s is true. The disjunction r https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif s is true. If x = 11, then r is false, and s is false. The disjunction r https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif s 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. p https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif q 12 is prime or 17 is prime. true 2. p https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif r 12 is prime or 19 is composite. false 3. q https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif r 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 a https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif b T T T T F T F T T F F F https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/tab.gif a b ~b a https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif ~b T T F T T F T T F T F F F F T T https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/tab.gif a b ~a ~a https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif b 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 x https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/and.gif y x https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif y 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 p https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif q. 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? Začátek formuláře (_) 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: _____________________________________________ Konec formuláře 2. Which of the following statements is a disjunction? Začátek formuláře (_) ~x https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/and.gif y (_) x https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/and.gif y (_) x https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif y (_) None of the above. RESULTS BOX: _____________________________________________ Konec formuláře 3. A disjunction is used with which connector? Začátek formuláře (_) And (_) Or (_) Not (_) None of the above. RESULTS BOX: _____________________________________________ Konec formuláře 4. If a is false and b is true, what is the truth value of a https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif ~b? Začátek formuláře (_) True (_) False (_) Not enough information was given (_) None of the above. RESULTS BOX: _____________________________________________ Konec formuláře 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? Začátek formuláře (_) r https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif ~s (_) r https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/and.gif ~s (_) r https://www.mathgoodies.com/sites/all/modules/custom/lessons/images/symbolic_logic/images/or.gif s (_) All of the above. RESULTS BOX: _____________________________________________ Konec formuláře