Mathematics for Economists
Chapter 1 : Mathematical foundations
By Lund University
In this chapter, we look at the foundations of mathematics. In particularly, we will look at logic and sets. You can go quite far in mathematics without the foundations. However, in order to develop a better understanding of mathematics, foundations are very useful as you will know the exact definitions of concepts we use in mathematics.
In this section, we explore very fundamental questions such as what exactly do we mean by a statement "x = 4". We will formally define fundamental terms such as predicates, implications and compound statement. We will also investigate various methods of constructing proofs in mathematics.
Statements and predicates
Compound statements and truth tables
Exercises on logic
Problem: Necessary and sufficient conditions
Problem: Implications and converse
Problem: if, only if, and if and only if
Problem: Implication and equivalence with squares
Problem: Proof that squares are non-negative
Problem: Proof with squares and inequalities
Problem: Proof with square and zero
Problem: Proof with sum of squares
Problem: Proof, cubes can be negative
Problem: Proof, fourth powers cannot be negative
Problem: Proof with squares and strict inequalities
Problem: Proof that cubic function is injective
This section is an introduction to an important topic in mathematics, namely set theory. Set theory is generally considered as a subtopic of mathematical logic. Informally, sets are collection of objects. We will introduce the basic notation, such as set membership and subsets. We will also look at some set algebra; union, intersection and complement. Finally, we look at Venn the diagram as a way of illustrating rules for set operations.
Exercises on sets