Fall 2002 CSCI 333 Homework 1

This homework is not graded. It is practice for the 4 September quiz.

Problem 1

For each of the following relations, prove if the relation is reflexive, symmetric, antisymmetric, and transitive. All of these relations are taken over the set of positive integers.

1. For integers a and b, a ≡ b if and only if a+b equals 10.
2. For integers a and b, a ≡ b if and only if a+b is greater than 10.
3. For integers a and b, a ≡ b if and only if a < b+3.
4. For integers a and b, a ≡ b if and only if a+b is divisible by 3.
5. For integers a and b, a ≡ b if and only if a equals 10.
6. For integers a and b, a ≡ b if and only if a*b < a+b.

Problem 2

Use induction to prove that the function defined by the following recurrence relation:

• T(0) = 1
• T(n) = T(n-1) + 2, for n ≥ 1

has the closed form solution

• T(n) = 2 n + 1