Assignment 3 (Proof Basics)

1. Following the style of theorems and proofs through Chapter 2, for each of the following statements, write a direct proof showing the statement is true, or give a counterexample.

   1. If x,y, and z are integers and x|(y+z) and x|y, then x|z.

   2. If x,y, and z are integers, x|z, and y|z, then and xy|z.

   3. if x and y are integers such that xy is a perfect square, then x and y are also perfect squares.

   4. If x is a rational number, then x² - 3x + 2 is also a rational number.

2. Highlighting the Need to Articulate Each Step in a Proof:
   This problem presents a proof that obviously must contain an error. However, on the surface, each step may seem plausible, so there is a need to carefully show and explain each step.

   Review the following "Theorem" and "Proof", and explain where the argument goes wrong—or is it actually true that 0 = 1?

   Theorem: 0 = 1
   Proof: We begin defining x = 1, and then present steps that show x also = 0.

   Step Number	Algebraic Step	Reason
   1	x = 1	Given
   2	x2 = x	Multiplying both sides by x
   3	x2 - 1 = x - 1	Subtracting 1 from both sizes
   4	(x-1)(x+1) = x - 1	Factoring
   5	x+1 = 1	Dividing by x-1
   6	x = 0	Subtracting 1 from both sides
   Conclusion:	Since x = 1 (Step 1) and x = 0 (Step 6), we conclude 1 = 0

created October 25, 2025 revised October-November, 2025
For more information, please contact Henry M. Walker at walker@cs.grinnell.edu.