1. Understanding Set Notation Students will use mathematical symbols to describe sets.

2. Warm Up • Classify each of the following numbers. Write all classifications that apply (real, rational, irrational, integers, whole, natural) • -4 • 3.7 • π {Z, Q, R} {Q, R} {Irrational, R} Algebra Review

3. What is a Set? • A set is collection of items called elements. For example we could call the people in this class a set. • A subset is a set whose elements belong to another set. For example, subsets could be: “boys” , “girls” or “people that are taking band”. • If a set has no members we say that it is the empty set. In this class that could be people under the age of 10. Algebra Review

4. Describing a Set • There are several ways to describe a set. One way is with words. Two other ways use braces. • In describing a set, braces { } mean “a set of”. Then there are two different ways to describe the set inside the braces: • Roster notation: lists all the elements of the set, or uses dots to represent the missing members. So we could write the set of whole numbers less than 11 as: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} or {0, 1, 2, 3, . . ., 10} Algebra Review

5. Set-Builder Notation • Another way to describe a set is by using set-builder notation. • This method uses symbols. {x|x < 11 and x∈W} • The braces { } mean “the set of” • “x|” means “all numbers x such that” • Everything after the “|” describes the set. In this case it means x is less than 11 and is a whole number. Algebra Review

6. An Example • Describe the set of whole numbers less than 20 but greater than 12 using: • Set-Builder notation • Roster notation or {x|x < 20 and x > 12 and x∈W} {13, 14, 15, 16, 17, 18, 19} {13, 14, 15, . . . 19} Algebra Review

7. The Language of Sets • If the elements of a set can be counted, we say this set is finite. • If the number of members of a set continues without end, then we say this set is infinite. • Is the set of all whole numbers less than 100 an infinite or finite set? • Is the set of all real numbers between 2 and 4 an infinite or finite set? finite infinite Algebra Review

8. | | | | | | | -2 -1 0 1 2 3 4 How do we graph real numbers? • Since rational and irrational numbers make up all the other numbers on the number line, we graph the set of real numbers like this: • We fill in all the space since real numbers represent all the numbers on the number line. Algebra Review

9. Interval Notation • Another way to describe a set of real numbers (and this should only be used for real numbers) is called interval notation. • In interval notation, the beginning point and the end point are listed inside parenthesis ( ) or brackets [ ]. • This method is used to describe an interval on a number line. • If the interval is infinite (goes on forever) then we use the symbol ∞. Algebra Review

10. | | | | | | | -2 -1 0 1 2 3 4 | | | | | | | -2 -1 0 1 2 3 4 Interval Notation • Here are some examples of interval notation The description of this set using interval notation is: The description of this set using interval notation is: [-1, 3] - the brackets mean that the numbers are included. (-1, 2) - the parentheses mean that the numbers are not included. Algebra Review

11. | | | | | | | -2 -1 0 1 2 3 4 | | | | | | | -2 -1 0 1 2 3 4 Interval Notation • Interval notation with an infinite interval The description of this set using interval notation is: The description of this set using interval notation is: (-∞, 3] The 3 is included but infinity can never be reached - always has a parenthesis. (2, ∞) Algebra Review