Definition of Sets

Text Box:

Introduction to Set Theory

Interactive Lesson by Thomas Cordell

Text Box: What are Sets? When talking about objects in mathematics, we most commonly talk about numbers. Sometimes these objects can be geometrical in nature or “structures”. No matter what they are, in mathematics the “objects” are called “elements”. Sometimes we wish to talk about more than one element or “collections”. In mathematics, collections of elements are called sets. For additional information, check out the Set Wikipedia. Examples of sets: A := {1, 2, 3} B := {red, white, blue} Important Properties of Sets: 1. The order in which elements of a set are listed is irrelevant: {1, 2, 3} = {1, 3, 2} = {2, 1, 3} = {2, 3, 1} = {3, 1, 2} = {3, 2, 1} 2. Sets are notated using capital letters (A, B, F, Z, etc) followed by a semi colon and an equal sign. For example, A := 3. Elements of sets are dictated using lower case letters. (a, b, f, z, etc) 4. The symbol Î means “in” 5. The symbol Ï means “not in”. 6. The symbol means all Natural numbers, which is 0 thru infinity 7. The symbol | means “such that” 8. All logical operators such as <, £, >, ³, =, and can be applied to sets. Examples of set notation: If set A contains the elements 5, 11, 17, and 18, we denote this set by: A := {5, 11, 17, 18} If element d belongs to a set D, we say that d is an element of D or that d is an element in D. In this case, we write: d Î D If element d is not an element of set D, we write: d Ï D If element n belongs to a set of all natural numbers and is not equal to 1 and is less than 120, we write: Conclusion: You have now learned the basics of how to notate mathematical sets using symbols and vice versa. Next we will move on to see sets in action. This next section will cover subsets, unions, symmetric difference, and intersections of sets.

Definition of Sets