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Discrete Mathematics: An Open Introduction, 4th edition (preview)

Oscar Levin

Worksheet Preview Activity
A4US

Before reading on to the main content of the section, complete this preview activity to start thinking about the types of questions this section will address.
1.
    Which of the following sentences should count as statements? That is, for which of the sentences below, could you potentially claim the sentence was either true or false? Select all that apply.
  • The sum of the first 100 positive integers.
  • This is not a statement. It is not even a complete sentence (there is no verb).
  • What is the sum of the first 100 positive integers?
  • This is a question. It is not a statement.
  • The sum of the first 100 positive integers is 5050.
  • This is a statement. It is either true or false (it happens to be true).
  • Is the sum of the first 100 positive integers 5050?
  • This is a question. The answer happens to be “yes”, but that is not the same as saying “true”. Questions are never statements.
  • The sum of the first 100 positive integers is 17.
  • This is clearly false. But since it is false, it is a statement!
2.
    Consider the statement, “If I see a movie, then I eat popcorn” (which happens to be true). Based solely on your intuition of English, which of the following statements mean the same thing? Select all that apply.
  • If I eat popcorn, then I see a movie.
  • This is not equivalent to the original statement. Maybe I also eat popcorn when I watch TV? In that case, the original statement would be true, but this one would be false.
  • If I don’t eat popcorn, then I don’t see a movie.
  • Correct.
  • It is necessary that I eat popcorn when I see a movie.
  • This is equivalent to the original statement (although here “necessary” is used in a logical sense).
  • To see a movie, it is sufficient for me to eat popcorn.
  • Just because I eat popcorn, doesn’t mean I see a movie. I might eat popcorn in other situations. So this is not equivalent to the original statement.
  • I only watch a movie if I eat popcorn.
  • Another way of saying this is, “I watch a movie only if I eat popcorn.” This is equivalent to the original statement.
3.
    Suppose that your shady uncle offers you the following deal: If you loan him your car, then he will bring you tacos. In which of the following situations would it be fair to say that your uncle is a liar (i.e., that his statement was false)? Select all that apply.
  • You loan him your car. He brings you tacos.
  • You loan him your car. He never buys you tacos.
  • You don’t loan him your car. He still brings you tacos.
  • Maybe he just really likes giving you tacos. That’s not enough to say he was a liar, is it?
  • You don’t loan him your car. He never brings you tacos.
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