Virtual Field Trip: We’ll visit the Academic Support Center, click “My Studies” on your KU Campus Homepage. The pdf and additional materials about Sets can be found at: http://www.ramshillfarm.com/Math/Math150/Unit_2.html Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

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KU Math Center Sunday, Wednesday & Thursday: 8:00pm-12:00am (midnight) Monday: 11:00am-5:00pm AND Tuesday: 11:00am-12:00am (midnight) * All times are Eastern Time Additional Information about the Math Center is in the Doc Sharing Portion of the course. Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Page 68 2.1 Set Concepts Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Set A set is a collection of objects, which are called elements or members of the set. The symbol , read “is an element of,” is used to indicate membership in a set. The symbol , means “is not an element of.” Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Well-defined Set A set which has no question about what elements should be included. Its elements can be clearly determined. No opinion is associated with the members. Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Roster Form This is the form of the set where the elements are all listed, each separated by commas. Description: Set N is the set of all natural numbers less than or equal to 25. Solution: N = {1, 2, 3, 4, 5,…, 25} The 25 after the ellipsis indicates that the elements continue up to and including the number 25. Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Set-Builder Notation A formal statement that describes the members of a set is written between the braces. A variable may represent any one of the members of the set. |, on the “\” key, is used to denote “such that”. Description: Set N is the set of all natural numbers less than or equal to 25. Solution: { x | x є N and 1 ≤ x ≤ 25} Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Cardinal Number Page 71 The number of elements in set A is its cardinal number. Symbol: n(A) A = { 1, 2, 3, 4, 6, 8} n(A) = 6 Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Empty (or Null) Set A null set (or empty set ) contains absolutely no elements, and so its cardinal number is 0. Symbol:

Finite Set A finite set is either empty or the cardinal number is finite. Example: Set S = {2, 3, 4, 5, 6, 7} is a finite set because the number of elements in the set is 6, and 6 is a natural number. Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Equivalent Sets Equivalent sets have the same cardinal number. Symbol: n(A) = n(B) A = { 1, 3, 5, 7, 9} B = { 2, 4, 6, 8, 10} A & B are equivalent Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Equal Sets Equal sets have the exact same elements in them, regardless of their order. Symbol: A = B Example: A = { 1, 2, 4, 5} B = { 2, 5, 4, 1} Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Page 77 2.2 Subsets Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Subsets A set is a subset of a given set, if everything in the subset is comes from the given set. Symbol: A B To show that set A is not a subset of set B, one must find at least one element of set A that is not an element of set B. The symbol for “not a subset of” is .

Determining Subsets Example: Determine whether set A is a subset of set B. A = { 3, 5, 6, 8 } B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Solution: All of the elements of set A are contained in set B, so A B. Note: B is a subset of itself! Í Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Proper Subset A set is a proper subset of a given set, if it is a subset of that set AND it is smaller than the given set. Symbol: or A set can be a Subset, but not a Proper Subset of itself.

Determining Proper Subsets Example: Determine whether the set A is a proper subset of the set B. A = { dog, cat } B = { dog, cat, bird, fish } Solution: All the elements of set A are contained in set B, and sets A and B are not equal, therefore A B. Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Determining Proper Subsets continued Determine whether the set A is a proper subset of the set B. A = { dog, bird, fish, cat } B = { dog, cat, bird, fish } Solution: All the elements of set A are contained in set B, but sets A and B are equal, therefore A B. Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Number of Distinct Subsets Page 79 The number of distinct subsets of a finite set A is 2n, where n = n(A), the cardinal number of A. Example: Determine the number of distinct subsets for the given set { t , a , p , e }. List all the distinct subsets for the given set: { t , a , p , e }. Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Number of Distinct Subsets continued Solution: Since there are 4 elements in the given set, the number of distinct subsets is 24 = 2 • 2 • 2 • 2 = 16 subsets {t,a,p,e}, {t,a,p}, {t,a,e}, {t,p,e}, {a,p,e}, {t,a}, {t,p}, {t,e}, {a,p}, {a,e}, {p,e}, {t}, {a}, {p}, {e}, { } Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

2.3 Venn Diagrams and Set Operations Page 83 Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Venn Diagrams A Venn diagram is a technique used for picturing set relationships. A rectangle usually represents the universal set, U. The items inside the rectangle may be divided into subsets of U and are represented by circles. Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Disjoint Sets Two sets which have no elements in common are said to be disjoint. The intersection of disjoint sets is the empty set. Disjoint sets A and B are drawn in this figure. There are no elements in common since there is no overlap- ping area of the two circles. Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Intersection The intersection of two given sets contains only those elements common to both of those sets. and generally means intersection Symbol: Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Union The union of two given sets contains all of the elements for those sets, excluding duplicates. or generally means union Symbol:

Complement of a Set The set known as the complement contains all the elements of the universal set, which are not listed in the given subset. Symbol: A´ Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Subsets When every element of B is also an element of A. Circle B is completely inside Circle A. U A B Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

The Relationship Between n(A U B), n(A), n(B), n(A ∩ B) To find the number of elements in the union of two sets A and B, we add the number of elements in set A and B and then subtract the number of elements in the intersection of the sets. n(A U B) = n(A) + n(B) – n(A ∩ B) Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Difference of Two Sets The difference of two sets A and B symbolized A – B, is the set of elements that belong to set A but not to set B. Region 1 represents the difference of the two sets. B A U I II III IV Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

2.4 Venn Diagrams with Three Sets And Verification of Equality of Sets Page 95 2.4 Venn Diagrams with Three Sets And Verification of Equality of Sets Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

General Procedure for Constructing Venn Diagrams with Three Sets page 96 Construct a Venn diagram illustrating the following sets. U = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 2, 5, 8} B = {2, 4, 5} C = {1, 3, 5, 8} U A B C V I III VII VI IV VIII II Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.

Example: Constructing a Venn diagram for Three Sets completed B C V I III VII VI IV VIII II 4 2 5 1,8 3 6,7 Can't Type? press F11 or F5 Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them.