## 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

#### Proofs

## 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

## Sets: Problems

Exercises on sets