1. Quiz – last week on the month both Algebra and Geometry Algebra range – Permutation, Combinations, Systems of Linear Inequalities, Linear Programming

2. Sets and Venn Diagrams page 204 A set is a list of objects in no particular order; they could be numbers, letters or even words. { ㄱ, ㄴ, ㄷ, ㄹ }, {2,3,5,7}, {red, yellow, green} A Venn diagram is a way of representing sets visually.

4. Properties of Sets An object is an element of a set when it is contained in the set. ( 원소 ) The universe (usually represented as U) is a set containing all possible elements, ( 전체집합 ) while the empty set or null set (represented as Ø or { }) is a set containing no elements. ( 공집합 ) The cardinality of a set is the number of elements in the set. A set with 3 elements has cardinality of 3, |P|=3 or n(P)=3 ( 원소의 개수 )

5. The complement ( 여집합 ) of a set is the set containing all elements of the universe which are not elements of the original set.

6. To explain, we will start with an example where we use whole numbers from 1 to 10. We will define two sets taken from this group of numbers: Set A = the odd numbers = { 1, 3, 5, 7, 9 } Set B = the numbers which are 6 or more in the group = { 6, 7, 8, 9, 10 } Some numbers from our original group appear in both of these sets. Some only appear in one of the sets. Some of the original numbers don't appear in either of the two sets. We can represent these facts using a Venn diagram.

7. The two large circles represent the two sets. The numbers which appear in both sets are 7 and 9. These will go in the central section, because this is part of both circles. The numbers 1, 3 and 5 still need to be put in Set A, but not in Set B, so these go in the left section of the diagram. Similarly, the numbers 6, 8 and 10 are in Set B, but not in Set A, so will go in the right section of the diagram. The numbers 2 and 4 are not in either set, so will go outside the two circles. Set A = {1,3,5,7,9} Set B = {6,7,8,9,10}

8. The final Venn diagram looks like this: We can see that all ten original numbers appear in the diagram. The numbers in the left circle are Set A { 1, 3, 5, 7, 9 } The numbers in the right circle are Set B { 6, 7, 8, 9, 10 }

9. Intersection ∩ or ^ ( 교집합 ) The intersection of sets A and B is those elements which are in set A and set B. A diagram showing the intersection of A and B is on the left.

10. Union U ( 합집합 ) The union of sets A and B is those elements which are in set A or set B or both. A diagram showing the union of A and B is on the right.

11. The union of two sets is the set containing all elements of either A or B, including elements of both A and B. This operation is written as A U B. For example, {1,2,3} U {2,3,4} = {1,2,3,4}. The intersection of two sets is the set containing all elements that are in both A and B. This is written as A ∩ B. For example, {1,2,3} ∩ {2,3,4} = {2,3}.

12. The sum of the cardinalities of the intersection and union of two sets is equal to the sum of the cardinalities of the two sets. n(A ∩ B) + n(A U B) = n(A)+ n(B) n(AUB) = n(A) + n(B) – n(A∩B)

13. Example/Practice page 206 Work out the answer to each of these questions (a)Which numbers are in the union of A and B? (b) Which numbers are in the intersection of A and B? (a) -3,-1,0, 1,2,3,5,6,7,8,9,11 (b) 1,3,6

14. A set is a subset ( 부분집합 )of another when all the elements in the first set are contained in the second set. A ⊆B A set's proper subset ( 진부분집합 ) are all subsets except the set itself, A ⊂B. In Korea you do not use proper subset, so you can use A  B for any subset in our class! All sets are subsets of the universe. By definition, all sets are subsets of themselves and therefore, the null set is a subset of all sets. Two sets are equal if they are subsets of each other.

15. Power Set The power set ( 멱집합 )of a set S is the set of all subsets of S.setsubsets The cardinality of the power set of S given S is finite is equal to 2 n, where n is the cardinality of S.cardinality

16. Practice pages 208-210 Problems 1-7

20. Homework Pages 210-211 Exercises 1-7

24. Classwork Pages 212-213