Below are links to sample problems along with their proofs. Problems like these may appear on the written portion of the waiver exam.
Similar problems can be altered and simplified to make multiple choice problems.

Working through these problems is meant to help you prepare for the Waiver Exam.
There are many ways to test knowledge over a subject and so these sample problems are just that, samples.
Problems on the waiver exam may not necessarily look exactly like these however they will cover the same topics which are:

- Basic Sets
- Functions
- Injective, Surjective, and Bijective Functions
- Logic
- General Proofs
- Induction
- Even/Odd Proofs
- Real Analysis
- Indexed Sets
- Open and Closed Sets
- Bounded and Unbounded Sets
- Sequences

- Linear Algebra
- Vector Spaces
- Linear Maps

- Abstract Algebra
- Divisibility and Remainders
- Equivalence Relations and Modular Arithmetic