Here, you can find a (non-exhaustive!) list of the contents and take-aways of Chapter 0: Fundamentals of Mathematics. This list serves as an opportunity to assess both how thoroughly you should read the chapter before opening it for the first time, and how well you have managed to follow along once you have read it.
Chapter 0: Fundamentals of Mathematics discusses
- the formal basics of mathematical texts, that is, notation as well as statements and arguments
- mathematical implication and its relation to necessary, sufficient and equivalent conditions
- notation and central concepts of set theory
- the formal foundation of functions and key concepts and properties
- the univariate limit concept, both in the context of sequences and functions
Someone with profound knowledge of the contents of this chapter should
- be familiar with the central mathematical symbols
- know what quantifiers are and why they are useful
- be able to connect the concepts of necessity, sufficiency and equivalence to mathematical implication
- be familiar with the mathematical statement concept, including what validity and soundness refer to in this context
- know what a mathematical set is in the formal sense, and be able to handle central concepts of set theory, e.g. intervals and index sets, empty set and universal superset, etc.
- be familiar with relations between and operations on sets, e.g. subset, complement, union and intersection, set difference and power set, and be able to deal with these concepts in concrete applications
- be able to explain how the mathematical concept of relations is useful in thinking about functions formally
- be familiar with the building blocs of the function concept: domain, codomain, mapping rule and graph
- know about inverse functions and preimages, and the difference between these concepts
- know basic rules for differentiating common functions (e.g. chain rule, product rule)
- know the formal definition of the limit concept for both functions and sequences, and how continuity of a function relates to it
- be familiar with simple rules for the limit, and further with L’Hôpital’s Rule and the Sandwich Theorem
and be able to answer a number of related questions, including
- What is the value of mathematics for the economist profession?
- Can a logical statement be true if it is not meaningful?
- If is a necessary condition for , can be true when is violated?
- How can quantifying statements be negated (using for all/exists)?
- What is a pairwise disjoint sequence of sets?
- Can an object be contained in a set more than once?
- What is the set difference when and ?
- What is the difference between a subset and a proper subset?
- What is the graph of a function in the formal sense? How does it relate to the cartesian product of the function’s domain and codomain?
- When , what are the domain and codomain of ? What is the range? Can you draw the graph?
- If at , the right limit is not equal to the left limit for some function , can exist?