3.2 Truth Tables for Negation, Conjunction, and Disjunction 117 Solution a) Let p q : : 1 million is greater than 1 billion. 1 million is equal to 1 billion. The statement “1 million is greater than or equal to 1 billion” means that 1 million is greater than 1 billion or 1 million is equal to 1 billion. The compound statement can be expressed as ∨ p q. We know that both p and q are false statements, since 1 million is less than 1 billion and 1 million is not equal to 1 billion. We substitute F for p and F for q and evaluate the statement: ∨ ∨ p q F F F Therefore, the compound statement “1 million is greater than or equal to 1 billion” is a false statement. b) Let p q r : : : Montreal is in Canada. Beijing is in Mexico. Santiago is in Japan. Recall that the word but is used to express a conjunction. Thus, the compound statement, “Montreal is in Canada or Beijing is in Mexico, but Santiago is not in Japan” can be written in symbolic form as p q r ( ) . ∨ ∧ ∼ If necessary, we can consult the Internet to determine that since Montreal is in Canada, statement p is a true statement. Since Beijing is in China, not in Mexico, statement q is a false statement. And since Santiago is in Chile, not in Japan, statement r is also a false statement. To evaluate the compound statement, p q r ( ) , ∨ ∧ ∼ we substitute T for p, F for q, and F for r and then simplify. ∨ ∧ ∼ ∨ ∧ ∼ ∧ p q r F ( ) (T ) F T T T Therefore, the original statement is a true statement. 7 Now try Exercise 51 Example 10 Subscription Streaming Services Worldwide The number of worldwide subscribers for the most subscribed streaming services in 2022 is shown in Fig. 3.1. Use this graph to determine the truth value of the following statement: There are more subscribers to Disney+ than Prime and there are more subscribers to Spotify than Apple Music, or the most subscribed service is not Netflix. Streaming Services Subscriptions in the World (millions) Other 185 Max 85 Apple Music 80 Netflix 225 Prime 205 Spotify 180 Disney+ 130 Figure 3.1 Source: AllTopEverything.com
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