MTH104 Sets and Logic Handouts

Topic no 01 · Introduction · Introduction

This course, MTH104: Sets and Logic, introduces two fundamental pillars of modern mathematics: Set Theory and Mathematical Logic. These two areas form the foundation upon which all of mathematics is built.

Why do we study it?

Understanding sets and logic is essential for: Developing rigorous mathematical thinking Understanding how mathematical proofs are constructed Building a solid foundation for advanced mathematics (calculus, algebra, analysis, etc.) Learning to reason clearly and avoid logical fallacies

What is the relation between sets and logic?

Set theory provides the language of mathematics – almost every mathematical object can be defined as a set. Logic provides the rules of reasoning – the framework for constructing valid arguments and proofs. Together, they form the foundation of mathematics: set theory gives us objects to work with, and logic gives us rules to reason about them.

Proof and Mathematical Logic

In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. Proof is a logical argument, not an empirical one. Key terminology: Theorem: A proposition that has been proved true in all cases. Conjecture: An unproven proposition for which there is some sort of empirical evidence. Mathematical Logic: The framework upon which rigorous proofs are built. It is the study of the principles and criteria of valid inference and demonstrations. To prove a statement is a theorem, one must demonstrate that it holds in all cases, not just in a few examples. Mathematical logic provides the tools to ensure that proofs are valid and conclusions follow correctly from premises.

Set Theory

In set theory, we study sets, which are informally collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. Set theorists explore what is provable—and what is not provable—when one is allowed to use all of the usual tools of mathematical constructions and proofs.

Tools of Set Theory

Set theorists use various mathematical tools to construct and analyze sets, including: Function spaces: Sets of functions between sets Completions: Adding limit points to spaces (like completing rational numbers to real numbers) Quotients: Forming new sets by identifying elements (like equivalence classes) Products: Cartesian products of sets (ordered tuples) Induction and recursion: Defining and proving properties of infinite structures

Summary of Topic 01: Introduction

This course covers the foundations of mathematics: Sets and Logic. A proof is a logical argument demonstrating a statement must be true. A theorem is a proved proposition; a conjecture is an unproven proposition. Mathematical logic provides the rules for valid reasoning and proof construction. Set theory studies collections of objects and forms the language of modern

Set theorists use tools like function spaces, completions, quotients, products, induction, and recursion. Set theory and logic together form the foundation upon which all of mathematics is built.

Topic no 02 · Basic Concepts of Sets · Sets and Elements

A fundamental concept in all branches of mathematics is that of a set. A set is any well-defined list, collection, or class of objects. The objects in sets can be anything: numbers, people, letters, rivers, etc. These objects are called the elements or members of the set.

Examples of Sets:

1. The numbers 1, 3, 7 and 10.
2. The solution of the equation x² - 3x + 2 = 0.
3. The vowels of the English alphabet: a, e, i, o, u.
4. The people living on the earth.
5. The even integers 2, 4, 6, ...

Notation

Sets will usually be denoted by capital letters: A, B, X, Y, ... Elements in sets will usually be represented by lower case letters: a, b, x, y, ... The statement "p belongs to A" is written as p ∈ A. The statement "p is not an element of A" is written as p ∉ A.

Tabular Form (Roster Form)

If we define a particular set by actually listing its members, the elements are separated by commas and enclosed in curly brackets { }. This is called the tabular form of a set.

Example:

Let A consist of the numbers 1, 3, 7 and 10, then we write: A = {1, 3, 7, 10}

Set-Builder Form

If we define a particular set by stating properties which its elements must satisfy, we use a letter (usually x) to represent an arbitrary element and write:

B = { x | x satisfies property P }

This is called the set-builder form of a set.

Example:

Let B be the set of all even numbers. Then we write: B = { x | x is even } This is read as: "B is the set of all x such that x is even."

Summary of Set Representations:

Tabular Form: Listing all elements inside { }. Example: {1, 2, 3} Set-Builder Form: Describing elements by a property. Example: {x | x is a positive integer}

Topic no 03 · Empty Set, Subsets and Universal Set · Universal Set

All sets under investigation in any application of set theory are assumed to be contained in some large fixed set called the universal set or universe. We will denote the universal set by U.