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Module description - Foundation in Mathematics
(Mathematische Grundlagen)

ECTS 3.0
Level Basic
Content Mathematics forms an important basis for understanding the world of data. The main goal of this competency is to promote mathematical thinking so that practical applications from Data Science can be modeled using mathematics. The second goal is to learn about and use important mathematical tools that are needed in other competencies in the study to solve the modeled problems.
Learning outcomes Set Theory
Students understand the definition of a set and know the most important set operations, their properties and can calculate with sets. They can represent sets in different ways. The students know important examples of sets.

Students understand what a statement is, know the most important logical operators and can apply the most important laws of Boolean algebra to logical formulas. They can visualize logical formulas, evaluate them and put them into normal forms. They know the satisfiability problem (SAT).
They understand the predicates and can form correct statements from predicates using quantifiers. Students can translate statements from a natural language into the language of mathematics using logical operators and quantifiers. They can prove or disprove statements.

Proof Techniques
Students understand direct and indirect proofs as an application of logic and can apply it to simple examples. They understand the concept of mathematical Induction.

Relations / Functions
Students will be able to examine binary homogeneous relations for reflexivity, symmetry, antisymmetry, and transitivity, and will know graphical representations in addition to set representations.
Students know equivalence relations. They understand the connection to partitions. They know partial orders as an order relation and can represent it graphically by the the Hasse diagram.
Students understand functions as a special case of relations. They know the explicit representation and the representation as a function graph. They know when the composition of two functions and the inverse of a function are defined and can determine them.

Students know graphs and their graphical representation. They know the terms node degree, subgraph and connectedness. With the help of Hierholzer's algorithm, they can determine Euler paths and Euler circles.
They understand trees as special cases of graphs. They can use algorithms to determine minimal spanning trees of given graphs.

Formal languages
Students know the concepts of an alphabet, of a word and of a formal language. They can perform some simple operations on words.
As a first simple computational model, they learn about finite automata. They can design finite automata for regular languages in the graphical representation.
As a second mechanism for solving the regular language decision problem, they define regular expressions. They understand the syntax and semantics of these and know the most important calculation rules.
The third mechanism for regular languages are regular grammars. In turn, they understand the syntax and semantics of them and can design simple grammars.
Evaluation Mark
Modultype Basic Module
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