ITEC 122 2008fall ibarland

home—info—lectures—exams—archive

hw04
Proof Strategies

1. Rosen p85, #8 Prove that if n is a perfect square, then n+2 is not a perfect square.
   You'll certainly need a definition of “perfect square”; see Rosen p.77, Example 2.
   You may use (as a justification) the following: Theorem: If ∃k.k² < m < (k+1)², then m is not a perfect square.
2. Rosen p85, #11 variant (= Rosen 5ed: p.75 #27 variant): Prove or disprove that the sum of two irrational numbers must be irrational.
3. Rosen: p85, #12 (= Rosen 5ed: p.75 #28): (Q-{0})·(ℜ-Q) ∈ (ℜ-Q)?
4. Rosen: p85, #22 Show that if you pick three socks from a drawer having only blue and black socks, you must get either a pair of blue socks or a pair of black socks.
5. Rosen: p102, #7 variant (= Rosen 5ed: p.76 #49 variant): Show that there are 5000 consecutive positive integers which are not perfect squares.
   A point of extra credit: instead, show that for any non-negative integer n, there are n consecutive positive integers which are not perfect squares.
6. Rosen: p102, #10 (= Rosen 5ed: p.76 #52): a pair with a nonnegative product
   A point of extra credit: instead, show that for any three real numbers a, b, c the product of two of them is non-negative.
7. Rosen: p102, #12: Prove or disprove: a rational number raised to a rational number is rational. “rationals closed under exponentiation?”
   Hint: Recall that √2 = 2^1/2.

home—info—lectures—exams—archive

©2008, Ian Barland, Radford University Last modified 2008.Oct.13 (Mon)

Please mail any suggestions (incl. typos, broken links) to iba�rlandrad�ford.edu