By Edward A. Bender

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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.

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. Write the statement, “ ∃! x ∈ D ∋ S(x)” using ∀ and ∃ instead of ∃!. 18. In each case, is the given statement true or false?

Is the statement form (p ∧ ∼q) ∧ (∼p ∨ q) ∧ r a tautology, contradiction, or neither? 7. Is the statement form (∼p ∧ q) ∧ (q ∨ r) ∧ ∼q ∧ r a tautology, contradiction, or neither? 8. Construct a truth table for p ∨ (∼p ∧ q) ⇒ q. 9. Construct a truth table for p ∨ (∼p ∧ q) ⇒ ∼q. 10. Construct a truth table for (p ⇒ q) ⇒ (q ⇒ p). 11. Write negations of the following statements in English that is as easily understood as possible. (a) If P is a pentagon then P is a polygon. (b) If Tom is Ann’s father, then Jim is Ann’s uncle and Sue is her aunt and Mary is her cousin.