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

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} FHS Algebra Review

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. FHS Algebra Review

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} FHS Algebra Review

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. FHS Algebra Review

{x|x < 20 and x > 12 and x∈W} 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} FHS Algebra Review

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 FHS Algebra Review

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. | | | | | | | -2 -1 0 1 2 3 4 FHS Algebra Review

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 ∞. FHS Algebra Review

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

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