Mathematics Foundations/1.1 Basic Concepts and Notation

Introduction Set theory forms the foundation of modern mathematics. A set is simply a collection of distinct objects, considered as an object in its own right. The objects that belong to a set are called its elements or members. This section introduces the fundamental concepts and notation of set theory that will be used throughout this textbook. Definition of a Set A set is a well-defined collection of distinct objects. By "well-defined," we mean that it is possible to determine whether a given object belongs to the collection or not. The objects in a set are called elements or members of the set. If an object is an element of a set , we write , if is not an element of , we write Set Notation There are several ways to denote sets: 1. Roster Notation (Listing Method) In roster notation, we list all elements of the set inside curly braces, separated by commas: This defines a set A containing the elements 1, 2, 3, 4, and 5. If a set contains a pattern that is clear, we can use ellipses: This defines a set containing all integers from 1 to 100. 2. Set-Builder Notation In set-builder notation, we define a set by stating a property that its elements must satisfy: This defines a set containing all even integers between 2 and 10 inclusive: . The vertical bar is read as "such that," and the expression can be interpreted as "the set of all such that is an even integer and ." Special Sets Several sets are so important in mathematics that they have standard symbols: or : The empty set (or null set), which contains no elements : The set of all natural numbers: : The set of all integers: : The set of all rational numbers (fractions) : The set of all real numbers : The set of all complex numbers Cardinality The cardinality of a set A, denoted by or , is the number of elements in the set. Examples: If , then If (the empty set), then Subsets A set A is a subset of a set B, written as A⊆B, if every element of A is also an element of B. If A⊆B and A=B (i.e., there is at least one element in B that is not in A), then A is a proper subset of B, written as A⊂B. Examples: If and , then If and , then but not Power Set The power set of a set A, denoted by or , is the set of all subsets of , including the empty set and itself. Example: If , then: = Note that if then . Universal Set In a particular context or discussion, the universal set is the set of all elements under consideration. All other sets in the discussion are subsets of . Venn Diagrams Venn diagrams are graphical representations of sets. In a Venn diagram: The universal set is represented by a rectangle Sets are represented by circles (or other closed curves) within the rectangle Elements are represented by points within these circles Venn diagrams are particularly useful for visualizing relationships between sets, as we will see in the next section on set operations. Examples Example 1: . List all elements of and find Solution: A consists of all integers from -3 to 4: . Therefore, . Example 2: Let . List all elements of and find . Solution: The power set of consists of all possible subsets of : Since , we have . Exercises Let . Determine whether each statement is true or false: a) b) c) d) e) 2.Write the following sets using roster notation: a) b) c) 3.Let . List all elements of and . 4.If a set has 5 elements, how many elements does its power set have? 5.Determine whether each of the following collections is a well-defined set: a) The set of all tall people b) The set of all integers greater than 10 c) The set of all beautiful mathematical formulas d) The set of all prime numbers between 10 and 30. Historical note Modern set theory was developed by Georg Cantor (1845-1918) in the late 19th century. His work on infinite sets and cardinality revolutionized mathematics and provided a foundation for many branches of mathematics. However, early set theory encountered paradoxes, such as Russell's paradox, which led to the development of axiomatic set theory. The most widely accepted axiomatic system is Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), which we will explore in Section 1.5.