Assignment 4 (Sets and More Proofs)

1. Following the style of theorems and proofs through Chapter 2, prove each of the following statements is true, or give a counterexample.

   1. If n and m are integers such that n² + m² is odd, then m is odd or n is odd.

   2. If n and m are integers, such that either m is odd or n is odd, then n² + m² is odd

   3. Prove that among any group of 1000 people, at least three of the people have the same birthday.
      Hint: Although not required in your work, you may consider proving this result by contradiction.

2. Recall the following three definitions for Cartesian productions and sets.

   • Definition of Cartesian product: (x, y) ∈ A ⨯ B ↔ (x ∈ A) ∧ (y ∈ B)
   • Definition of intersection: x ∈ A ∩ B ↔ (x ∈ A) ∧ (x ∈ B)
   • Definition of union: x ∈ A ∪ B ↔ (x ∈ A) ∨ (x ∈ B)

   Note: In the textbook and in the above definitions, the ↔ symbol (or "biconditional" symbol) is used to denote, "if and only if".

   1. Give a formal proof for the identity: A ⨯ (B ∩ C) = (A ⨯ B) ∩ (A ⨯ C)

   2. If set A has m elements and set B has n elements, how many elements are there in A ⨯ B? Explain briefly.
      Note: Depending on your background and experience, this problem may or may not seem straightforward. In any case, in a section of CS 415 I taught a couple years ago, several students gave the significantly wrong result (when they were working on a more extensive problem), so be sure you are comfortable with the answer. (Please see me if you have questions!)

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