# Introduction to University Mathematics

**Lecturer:** Jokke Häsä, email: firstname.surname 'at' helsinki.fi

This course is taught in Finnish. The course assignments, course material and lectures are all in Finnish. However, you can submit your coursework in English. The exams are provided in English if needed. **If you would like to take the exam in English, please contact the lecturer.**

#### Literature

- Daniel J. Velleman: How To Prove It - A Structured Approach ( Helsinki university library has one copy and you can also buy this book as eBook for example here ).
- Richard Hammack: Book of Proof
- Kenneth H. Rosen: Discrete Mathematics and Its Applications.
- Complex numbers (sections 3.1-3.13)

#### Topics

Techniques of proof:

- Mathematical induction and strong induction (also called complete induction)
- Proofs involving sets: how to show that a set is subset of another set, how to show that two sets are equal
- Examples and counterexamples
- Proofs involving conditionals and biconditionals: how to prove a statement of the form “if …, then …”; how to prove a statement of the form “… if and only if…”
- Proof by contradiction

Sets:

- Notation
- Empty set
- Subset
- Operations on sets: union, intersection, difference
- Universe and complement
- Power set
- Ordered pairs and cartesian products of sets

Functions:

- Definition
- Image of a subset
- Image of a function
- Inverse image (also called preimage)
- Injection (one-to-one) and surjection (onto), bijection
- Inverse function

Relations:

- Definition (binary relation)
- Reflexive, symmetric and transitive relations
- Equivalence relations
- Equivalence classes

Complex numbers (for mathematics students)

- Definition
- Complex plane
- Basic arithmetic with complex numbers
- Polar form and exponential form of complex numbers
- Solving the quadratic equation
- Integer powers and roots of complex numbers

Mathematics for statistics and computer science students:

- Basic logic: Logical connectives and negations of them, existential and universal quantifiers and negations of them
- Geometric progression (also called geometric sequence) and geometric series
- Recursion
- Binomial coefficients
- Logarithms, especially binary logarithm
- Modular arithmetic (also called clock arithmetic)
- Graphs

#### Course assignments

Even though the problem sheets are in Finnish, you can probably understand many of the questions and you can perhaps translate them with Google translate. The problem sheets can be found under the title "Harjoitustehtävät ja niiden ratkaisuehdotukset" on the Finnish Website.