By Edward A. Bender

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**Extra resources for A Short Course in Discrete Mathematics**

**Example text**

A) Let P be the set of people. For x ∈ P , let L(x) indicate that x has a large income and H(x) that x is happy. Rewrite the given statement using the notation of logic rather than the English language. ” (c) Write the negation of the statement in logic notation. Move the negation inside the statement as far as possible. 16. Which of the following statements are true, which are false (∃! means “there exists exactly one”). (a) ∃! x ∈ Z ∋ 1/x ∈ Z. (b) ∀ x ∈ R ∃! y ∈ R ∋ x + y = 0. 17. Let S be a predicate with domain D.

To do so, we must face two problems: • It is not clear how we should view “If X, then Y ” when X is false. For example, what should we think if it isn’t raining? • Due to the variety and ambiguity of English, translation into Boolean statements may not be clear. In the remainder of this section, we investigate carefully the relationship between English language assertions and Boolean functions (Boolean statement forms) associated with implication. ” Usually, when such a statement is made we are primarily concerned with the situation when r is true.

Most mathematicians believe that his proof was incorrect. ” It can be used for the cases n = 3 and n = 4, but cannot be used in general. Some people believe Fermat assumed it would work in general because it worked in these two cases. This concludes our “number theory” examples. They were chosen to show you how to work with the notation of predicate logic. They were also chosen to introduce you to some famous problems in number theory. There are important applications of number theory to computer science, but not, so far as we know, applications of these particular examples.

### A Short Course in Discrete Mathematics by Edward A. Bender

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