{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Set 4.1 Questions

# Set 4.1 Questions - Exercise Set 4.1 H IS a In 1-3 u se the...

This preview shows pages 1–2. Sign up to view the full content.

Exercise Set 4.1 * In 1-3. use the definitions of even, odd, prime, and composite to justify each of your answers. 1. Assume that k is a particular integer. a. Is -17 an odd integer? b. Is 0 an even integer? c. Is 2k - 1 odd? 2. Assume that m and n are particular integers. a. Is 6m + 8n even? b. Is lOmn + 7 odd? c. If m > n > 0, is m 2 - n 2 composite? 3. Assume that rand 5 are particular integers. a. Is 4r 5 even? b. Is 6r + 45 2 + 3 odd? c. If rand 5 are both positive, is r 2 + 2r 5 + 52 composite? Prove the statements in 4-10. 4. There are integers m and n such that m > 1 and n > I and I 1. . ;; + -;; IS an mteger. 5. There are distinct integers m and n such that 1. + .! is an m n integer. 6. There are real numbers a and b such that .Ja + b = va +.Jb. 7. There is an integer n > 5 such that 2" - 1 is prime. 8. There is a real number x such that x > I and 2 x > x 10. Definition: An integer n is called a perfect square if, and only if, n = k 2 for some integer k. 9. There is a perfect square that can be written as a sum of two other perfect squares. 10. There is an integer n such that 2n 2 - 5n + 2 is prime. Disprove the statements in 11-13 by giving a counterexample. 11. For all real numbers a and b, if a < b then a 2 < b 2 . 12. For all integers n, if n is odd then n; I is odd. 13. For all integers m and n, if 2m + n is odd then m and n are both odd. In 14-16, determine whether the property is true for all integers, true for no integers, or true for some integers and false for other integers. Justify your answers. HIS. _a" = (_a)" 16. The average of any two odd integers is odd. Prove the statements in 17 and 18 by the method of exhaustion. 17. Every positive even integer less than 26 can be expressed as a sum of three or fewer perfect squares. (For instance, 10 = 1 2 + 3 2 and 16 = 4 2 .) 2 18. For each integer n with I ~ n ~ 10, n - n + II is a prime number. 19. a. Rewrite the following theorem in three different ways: as V , if __ then __ , as V __ , __ (with- out using the words if or then), and as If __ , then __ (without using an explicit universal quantifier).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}