This homework is not graded. It is practice for Quiz 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``R``b`if and only if`a`<`b`+333. - For integers
`a`and`b`,`a``R``b`if and only if`a`+333 <`b`. - For integers
`a`and`b`,`a``R``b`if and only if`a`equals 333. - For integers
`a`and`b`,`a``R``b`if and only if`a`+`b`equals 333.

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