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.

Logic: Lectures

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

Mathematical implication

Compound statements and truth tables


Logic: Problems

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

Sets: Lectures

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.


Sets: Problems

Exercises on sets

Problem: Set member, subset, union, intersection and difference

Problem: Sets and logic

Problem: Subset, union, intersection and difference

Problem: Union, intersection, difference and complement