Sets and Operations TSWBAT apply Venn diagrams in problem solving; use roster and set-builder notation; find the complement of a set; apply the set operations of intersection and union; identify a disjoint set

Think about it – How would you go about finding out this answer? Of 250 people surveyed, 175 have brown hair, 110 have brown eyes, and 85 have both brown hair and brown eyes. How many of these people have neither brown hair nor brown eyes?

Venn Diagrams Venn diagrams show relationships between sets Set A collection of objects, called elements Elements Numbers, letters, or physical objects They are written between braces {} Two sets are equal ONLY if they have the exact same elements

Drawing Venn Diagrams Universe Subsets The set of all elements being considered Illustrated by a rectangle Subsets A set contained within a set Denoted by the symbol Illustrated as circles within the rectangle U

Use a Venn diagram 110-85 175-85 85 90 25 U U 50 This blue rectangle is the universe U = 250

Different Notations Roster notation List the elements of a set in brackets Example The set of whole numbers greater than 10 {11, 12, 13, 14…} … or a ellipsis is used to show that the set continues without end following the established pattern

Different Notations Set-builder notation Describes the set by stating the properties that its elements must satisfy Example The set of whole numbers greater than 10 {x|x is a whole number and x > 10} Read as: the set of all x such that x is a whole number and x is greater than 10

Symbols used Є Read as “is an element of”, or “belongs to” 50 Є {x|x is a whole number and x>10} Read as “is not an element of” or “does not belong to” 7 Є {x|x is a whole number and x>10}

Empty Sets A set that has no elements is called and empty set or a null set It can be denoted two ways {} Ø Do NOT write 0, zero is a number and empty set does not contain zero as an element

Examples List the elements of each set {q|q is a whole number and q<0} {} {t|t is an integer and t<0} {…, -4, -3, -2, -1} {v|v Є the set of integers and v>0} {21, 22, 23…} {n|n Є the set of natural numbers and n<0} Ø

Classwork Complete numbers 1-4 on page 19 of your workbook Turn it in when you are finished

Complements When a subset is defined, its complement is also defined The complement is denoted by the’ symbol Set A Complement is A’ Complement The set of all elements in the universe (U) but do not belong to the set

Examples If U = {0, 1, 2, 3, 4 ] and A = {0, 1, 3}, what is the complement of A? A’ = {2, 4} U = {integers}, A = {0} {… -3, -2, -1, 1, 2, 3…} U = {real numbers}, A = Ø A’ = {real numbers}

Example U = {1, 2, 3}, A = {1, 2, 3} A’ = {} In your workbook complete page 19, 5-10

Operations with sets Intersection of sets U U U The elements BOTH sets contain Denoted with the symbol U U U

Example of Intersection B = {2, 5, 6, 7} A B = {2, 5} U

Operations with Sets Union of Sets U U U U All elements that are in either or both sets Denoted by the symbol U U U U

Example of Union A = {1, 2, 3, 4, 5} B = {2, 5, 6, 7} A U B = 1, 2, 3, 4, 5, 6, 7}

Examples A U B A C B U C A B C A (B U C) A U (B C) U U U U U A 20 8 7 9 U 4 2 3 B C

Classwork Complete WB page 20, 11-18