Sets and Functions: Understanding Sets and Their Relationships with Numbers, Study notes of Mathematics

Download Sets and Functions: Understanding Sets and Their Relationships with Numbers and more Study notes Mathematics in PDF only on Docsity! Haberman/Kling MTH 111c Section I: Sets and Functions Module 1: Sets and Numbers DEFINITION: A set is a collection of objects specified in a manner that enables one to determine if a given object is or is not in the set. In other words, a set is a well-defined collection of objects. EXAMPLE: Which of the following represent a set? a. The students registered for MTH 95 at PCC this quarter. b. The good students registered for MTH 95 at PCC this quarter. SOLUTIONS: a. This represents a set since it is “well defined”: We all know what it means to be registered for a class. b. This does NOT represent a set since it is not well defined: There are many different understandings of what it means to be a good student (get an A or pass the class or attend class or avoid falling asleep in class). EXAMPLE: Which of the following represent a set? a. All of the really big numbers. b. All the whole numbers between 3 and 10. SOLUTION: a. It should be obvious why this does NOT represent a set. (What does it mean to be a “big number”?) b. This represents a set. We can represent sets like b in roster notation (see box at top of next page). { }3 10 4, 5, 6, 7, 8, 9"All the whole numbers between and " = 2 Roster Notation involves listing the elements in a set within curly brackets: “{ }”. DEFINITION: An object in a set is called an element of the set. ( symbol: “∈”) EXAMPLE: 5 is an element of the set { }4, 5, 6, 7, 8, 9 . We can express this symbolically: { }5 4, 5, 6, 7, 8, 9∈ DEFINITION: Two sets are considered equal if they have the same elements. We used this definition earlier when we wrote: { }3 10 4, 5, 6, 7, 8, 9 ."All the whole numbers between and " = DEFINITION: A set S is a subset of a set T, denoted , if all elements of S are also elements of T. S T⊆ If S and T are sets and S T= , then . Sometimes it is useful to consider a subset S of a set T that is not equal to T. In such a case, we write and say that S is a proper subset of T. S T⊆ S T⊂ EXAMPLE: { }4,7, 8 is a subset of the set { }4, 5, 6, 7, 8, 9 . We can express this fact symbolically by { } { }4, 7, 8 4, 5, 6, 7, 8, 9⊆ . Since these two sets are not equal, { }4,7, 8 is a proper subset of { }4, 5, 6, 7, 8, 9 , so we can write { } { }4, 7, 8 4, 5, 6, 7, 8, 9⊂ . 5 Since we use the real numbers so often, we have special notation for subsets of the real numbers: interval notation. Interval notation involves square or round brackets. Use the examples below to understand how interval notation works. CLICK HERE FOR AN INTRODUCTION TO INTERVAL NOTATION EXAMPLE: { }2 3 [ 2, 3] We use square brackets here since the endpoints a Set-buil re include der Notation Interval Notation d x x x∈ − ≤ ≤ = ↑ ↑ −and R a. { }2 3 ( 2, 3) We use round brackets here since the endpoints are NOT includ Set-builder Notation Interval Notation ed. x x x∈ − < < = ↑ ↑ −and Rb. { }2 3 ( 2, 2 3] We use a round bracket on the left since is NOT included Set-builder Notation Interval Notation . x x x∈ < ↑ − = − ↑ − ≤and Rc. { }2 3 [ 2, 3 3) We use a round bracket on the right since is NOT included Set-builder Notation Interval Notation . x x x∈ − ≤ < = ↑ ↑ −andRd. 6 EXAMPLE: When the interval has no upper (or lower) bound, the symbol (or ) is used. ∞ −∞ { }4 ( , 4] We ALWAYS use a round bracket with since it is NOT a number in th Set-builder Interval Notation No e tati . n set o x x x −∈ ≤ = ↑ ↑ − ∞∞andR a. { }4 [4, ) We ALWAYS use a round bracket with since it is NOT a number in the Set-builder Interval Notation No set. tation x x x ∞∈ ∞ ↑ ↑ ≥ =andRb. CLICK HERE FOR SOME INTERVAL NOTATION EXAMPLES EXAMPLE: Simplify the following expressions. a. ( ) [ ]4, 8, 3− ∞ ∪ − b. ( )4, ( , 2]− ∞ ∪ −∞ c. ( )4, ( , 2]− ∞ ∩ −∞ d. ( ) [ ]4, 10, 5− ∞ ∩ − − SOLUTION: a. ( ) [ ]4, 8, 3 [ 8, )− ∞ ∪ − = − ∞ b. ( )4, ( , 2] ( , )− ∞ ∪ −∞ = −∞ ∞ = R c. ( )4, ( , 2] ( 4, 2]− ∞ ∩ −∞ = − 7 d. ( ) [ ]4, 10, 5 Ø− ∞ ∩ − − = CLICK HERE FOR A SUMMARY OF INTERSECTIONS, UNIONS, SET-BUILDER NOTATION AND INTERVAL NOTATION WITH NUMBER LINES