1. Working WITH Sets Section 3-5

2. Goals Goal To write sets and identify subsets. To find the complement of a set. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary Set Roster Form Set-Builder Notation Empty Set Venn Diagram Universal Set Complement of a Set Subset

4. Why Study Set Theory? Understanding set theory helps people to … see things in terms of systems organize things into groups begin to understand logic

5. Studying sets helps us categorize information. It allows us to make sense of a large amount of information by breaking it down into smaller groups. Set Concepts

6. A set is a collection of objects. –These objects can be anything: Letters, Shapes, People, Numbers, Desks, cars, etc. –Notation: Braces ‘{ }’, denote “The set of …” These objects are called elements or members of the set. The symbol for element is . For example, if you define the set as all the fruit found in my refrigerator, then apple and orange would be elements or members of that set. Sets:

7. Sets are inherently unordered: –No matter what objects a, b, and c denote, {a, b, c} = {a, c, b} = {b, a, c} = {b, c, a} = {c, a, b} = {c, b, a}. All elements are distinct (unequal); multiple listings make no difference! –{a, b, c} = {a, a, b, a, b, c, c, c, c}. –This set contains at most 3 elements!

8. There are three methods used to indicate a set: 1.Description 2.Roster form 3.Set-builder notation Venn Diagram - Used to display the contents of a set and the relationships between sets. Sets:

9. Description means just that, words describing what is included in a set. Example: “Set M is the set of months that start with the letter J.” 1. Description:

10. Roster form lists all of the elements in the set within braces {element 1, element 2, …}. Example: Set M = { January, June, July} 2. Roster Form:

11. Set-builder notation is frequently used in algebra. Example: M = { x  x  is a month of the year and x starts with the letter J} This is read, “Set M is the set of all the elements x such that x is a month of the year and x starts with the letter J”. 3. Set-Builder Notation:

12. In summary the three methods used to describe a set are: 1)Description: Set A is the integers 1, 2, 3, and 4. 2)Roster form: Set A = { 1, 2, 3, 4 } 3)Set-builder notation: –A = { x  x = 1, 2, 3, 4 } Set Summary:

13. Designating Sets Sets are commonly given names (capital letters). A = {1, 2, 3, 4} The set containing no elements is called the empty set (null set) and denoted by { } or

14. The Empty Set Any set that contains no elements is called the empty set the empty set is a subset of every set including itself notation: { } or  Examples ~ both A and B are empty A = {x | x is a Chevrolet Mustang} B = {x | x is a positive number  0}

15. Set Notation Elements an element is a member of a set notation:  means “is an element of”  means “is not an element of” Examples: –A = {1, 2, 3, 4} 1  A6  A 2  Az  A –B = {x | x is an even number  10} 2  B9  B 4  Bz  B

16. Give a complete listing of all of the elements of the set {x|x is a natural number between 3 and 8} Solution {4, 5, 6, 7} When listing the elements of a set, elements that occur more than once, are not repeated when listing the elements in set notation. Example: Listing Elements of Sets

17. Set Theory Notation Summary SymbolMeaning Upper casedesignates set name Lower casedesignates set elements { }enclose elements in set  or is (or is not) an element of | or :such that (if a condition is true)

18. VENN DIAGRAMS Venn diagrams are useful for presenting a visual picture of set relationships.

19. Venn Diagrams Sets can be represented graphically using Venn diagrams. In Venn diagrams: –A rectangle represents the universal set. –Circles (and other geometric figures) represents sets. –Points (or words, nunbers) represent elements.

20. Venn Diagrams

21. When working with a large group of information, we often break it into smaller sets called subsets. Subsets

22. Subsets of a Set Set A is a subset of set B if every element of A is also an element of B. In symbols this is written U A B

23. Set A is a subset of set B, symbolized by A  B, if and only if all the elements of set A are also elements of set B. So to be a subset, all elements of the set are also elements in another set (which is either the same size or larger than the first set). Subsets:

24. Subsets a subset part of or equal to another set notation:  means “is a subset of”  means “is not a subset of”

25. Given the sets A = { 1, 2 }, B = { 1, 2, 3 }, and D = { 1, 2, 3 } 1. A  B (said “A is a subset of B”) since all A is in B. Note that this cannot be written in reverse since B is not a subset of A. 2.D  B since all D is in B. Subset Examples:

26. Fill in the blank with to make a true statement. a) {a, b, c} ___ { a, c, d} b) {1, 2, 3, 4} ___ {1, 2, 3, 4} Solution a) {a, b, c} ___ { a, c, d} b) {1, 2, 3, 4} ___ {1, 2, 3, 4} Example: Subsets

27. 1.A  A (meaning - every set is a subset of itself). 2.The empty set, , is a subset of every set, including itself. Other Interesting Points About Subsets:

28. Number of Subsets The number of subsets of a set with n elements is 2 n.

29. Example: Number of Subsets Find the number of subsets of the set {m, a, t, h, y}. Solution Since there are 5 elements, the number of subsets is 2 5 = 32.

30. The number of distinct subsets of a finite set A is 2 n, where n is the number of elements in set. Example: Given the set { S,L,E,D }. The set has 4 elements, and 2 4 = 16. Thus, there are 16 distinct subsets for that set (note that the empty set is one of those 16 sets). One Last Point:

31. Two more important sets to consider are the empty set (also called null set) and the universal set. The empty set is the set that contains no elements. It is symbolized by { } or by . The universal set, symbolized by U, is the set of all elements for any specific discussion. More Sets:

32. Universal Set The universal set is the set of all things pertinent to a given discussion and is designated by the symbol U Example: U = {all students at ATC} Some Subsets: A = {all HS students} B = {freshmen students} C = {sophomore students}

33. Universal Set Example the universal set is a deck of ordinary playing cards each card is an element in the universal set some subsets are: –face cards –numbered cards –suits –poker hands

34. Universal Set U A In a venn diagram the rectangle represents the universal set, U, and it is required for all venn diagrams.

35. More on the Empty Set A set that has no elements is called the empty set or null set. Yes, it is still considered a real set, even though it has no elements. It is denoted by , or by { }. Since the empty set is a set, another set can contain the empty set as one of its elements: A ={ , a} This set has 2 elements B = {  }This set has 1 element C =  This set has 0 elements

36. If we are given the universal set U = { Chris, Tom, Alex }, then only these three names can be considered when working with the problem. If A = { x  x  U and x starts with the letter J}, then our answer would be the empty set (  ), since none of the names in our universal set start with the letter J. Empty Set and Universal Set – Example:

37. If U = { 1, 2, 3, 4, 5, 6 }, and A = { 1, 2, 3 }, then A = { ? }. If U = { 1, 2, 3, 4, 5, 6 }, and A = { 1, 2, 3 }, then A = { 4, 5, 6 }. The complement of a set A, symbolized by A, is all the elements in the universal set that are not in A (everything outside of A). Complement of a Set: One easy way to find this if the sets are in roster form is to cross out each element in U that is in set A. Then, whatever is not crossed out in U, is an element of A.

38. Complement of a Set: The complement of a set is the set of elements which do not belong to the set being complemented. Equivalent to the logic operation “not” Written as a prime, A ’, or a superscripted ‘c’, A c. Example: U = {a, b, c, d, e, u, v, w, x, y, z} A = {a, b, c, x, y, z} and B = {a, b, c, d, e} A ’ = {d, e, u, v, w} B c = {u, v, w, x, y, z}

39. Complement of Sets: Venn Diagrams U = {a, b, c, d, e, u, v, w, x, y, z} A = {a, b, c, x, y, z} B = {d, e, y, z} A ’ = {d, e, u, v, w}

40. Joke Time What is the best state to buy a new soccer uniform in? New Jersey Why is a football stadium always a cool place to sit? It’s full of fans! What did the pony say when he had a cold? I’m just a little horse!

41. Assignment 3-5 Exercises Pg. 213 – 215: #10 – 56 even