3.1 Statements and Logical Connectives 101 In Example 2, the given statement “Jon Batiste is the leader of the band Stay Human, but Jon Batiste is not playing the Chicago Jazz Festival” could also have been given as “Jon Batiste is the leader of the band Stay Human, but he is not playing the Chicago Jazz Festival.” It should be clear that the word he refers to Jon Batiste. Therefore, the statement “Jon Batiste is the leader of the band Stay Human, but he is not playing the Chicago Jazz Festival” can also be symbolized as l p. ∧ ∼ Or Statements The disjunction is symbolized by ∨ and read “or.” The or we use in this book (except where indicated in the exercise sets) is the inclusive or described on page 98. Profile in Mathematics George Boole The self-taught English mathe_ matician George Boole (1815– 1864) took the operations of algebra and used them to extend Aristotelian logic. He used symbols such as x and y to represent particular qualities or objects in question. For example, if x represents all butterflies, then − x (1 ) represents all things that are not butterflies. If y represents the color yellow, then − y (1 ) represents all colors except for yellow. Furthermore, − − x y (1 )(1 ) represents all things that are not butterflies and not yellow. This development added a computational dimension to logic that provided a basis for twentieth-century work in the field of computing. Example 3 Write a Disjunction Let p q : Camila will take chemistry. : Camila will take French. Write the following statements in symbolic form. a) Camila will take chemistry or Camila will take French. b) Camila will not take chemistry or Camila will not take French. c) Camila will take French or Camila will not take chemistry. Solution a) ∨ p q b) p q ∼ ∨ ∼ c) q p ∨ ∼ Now try Exercise 29 7 Because or represents the inclusive or , the statement “Camila will take chemistry or Camila will take French” in Example 3(a) may mean that Camila will take chemistry, or that Camila will take French, or that Camila will take both chemistry and French. The statement in Example 3(a) could also be written as “Camila will take chemistry or French.” When a compound statement contains more than one connective, a comma can be used to indicate which simple statements are to be grouped together. When we write the compound statement symbolically, the simple statements on the same side of the comma are to be grouped together within parentheses . For example, “Harry Styles is a singer s( ) or Zendaya is an actor z( ), and Dallas is in Texas d( )” is written s z d ( ) . ∨ ∧ Note that the s and z are both on the same side of the comma in the written statement. They are therefore grouped together within parentheses. The statement “Harry Styles is a singer, or Zendaya is an actor and Dallas is in Texas” is written s z d ( ). ∨ ∧ In this case, z and d are on the same side of the comma and are therefore grouped together within parentheses. Example 4 Understand How Commas Are Used to Group Statements Let p q r : Dinner includes soup. : Dinner includes salad. : Dinner includes the vegetable of the day. Write the following statements in symbolic form and indicate whether the statement is a negation, conjunction, or disjunction. a) Dinner includes soup, and salad or the vegetable of the day. b) Dinner includes soup and salad, or the vegetable of the day.
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