# 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.

- For integers
`a` and `b`,
`a` ≡ `b` if and only if
`a`+`b` equals 10.
- For integers
`a` and `b`,
`a` ≡ `b` if and only if
`a`+`b` is greater than 10.
- For integers
`a` and `b`,
`a` ≡ `b` if and only if
`a` < `b`+3.
- For integers
`a` and `b`,
`a` ≡ `b` if and only if
`a`+`b` is divisible by 3.
- For integers
`a` and `b`,
`a` ≡ `b` if and only if
`a` equals 10.
- 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