Module 6 Homework Problems

Complete the following problems…

… and be patient, taking your time! Some of these problems are challenging! Challenging problems usually are more important and worth remembering! Note also that the statement or proof of one problem can certainly be used to prove another statement! That’s part of the beauty of mathematics; it builds on itself!

IF YOU NEED HELP OR WANT A HINT, PLEASE ASK!!!!

Prove or Disprove the following statements (i.e. prove them if you think they are true; disprove if you think they are false).

(Hint 1: If a proof isn’t working out, consider the possibility that the statement given might be false)

(Hint 2: Why you think a statement is true tends to indicate how to prove the statement)

1. “If \(n\) is odd, then \(\frac{(n+3)}{2}\) is even”
2. “If \(n\) is a natural number, then there are at least three natural numbers smaller than \(n\).”
3. “The sum of any two consecutive natural numbers is odd”
   • Natural numbers \(n\) and \(m\) are “consecutive” if \(m=n+1\) or vice versa. That is, one of the numbers is one more than the other.
4. “The sum of any three consecutive natural numbers is divisible by \(3\).”
   • Ask for help by email if this is unclear or if you’re not sure where to start!
5. (Bonus +10) “If \(a\) and \(b\) are real numbers, then there exists another real number \(c\) such that \(a\leq c\leq b\)”
   • This is an important idea! See if you can express it in plain English, and remember it.

Prove the following statements by contraposition (or directly if you prefer or are able to)

6. “If \(n^2\) is odd, then \(n\) is odd.”
7. “If \(n^3\) is even, then \(n\) is even.”
8. “If \(n^2\) is divisible by \(3\), then \(n\) is divisible by \(3\).”
   • Hint: When a number \(m\) is NOT divisible by \(3\), the Quotient Remainder Theorem gives us that either \(m=3k+1\) or \(m=3k+2\) for some integer \(k\) (i.e. \(m\) has either a remainder of \(1\) or \(2\) when divided by 3). In either case, see what happens to \(m^2\).
9. Let \(r,s\in \mathbb{R}\). “If \(r+s<100\), then \(r<50\) or \(s<50\).”
10. Let \(r\in \mathbb{R}\). “If \(\frac{1}{r}\) is irrational, then \(r\) is irrational.”

Prove the following statements by contradiction (or directly, if you can or prefer)

11. “The difference of a rational number and an irrational number is irrational.”
12. “The product of an irrational number and a rational number is irrational.”
13. “There is no smallest positive real number.”
    • This is a really important idea, and worth remembering!
14. “If \(n\) is a rational number, then \(n\cdot \sqrt{2}\) is irrational.”
    1. Hint: This statement is a corollary of a prior statement. Figure out which one and use it in your proof of this statement!
15. (Show why this statement is false) “The product of two irrational numbers is irrational”
16. “\(\sqrt{3}\) is irrational”
    1. Hint: There is a statement you proved earlier that can be used as a lemma here.
17. (Bonus +20) “If \(p\) is prime, then \(\sqrt{p}\) is irrational”
    1. You may use the fact that “If \(p\vert n^2\), then \(p\vert n\)” without needing to prove it.
    2. Note: This theorem is important and worth remembering!
18. (Bonus +10) “If \(n\) is an integer, then \(n^2-2\) is not divisible by \(4\)”