2. Discrete Unit 1 Lesson Plan 1) SWDAT use the properties and symbols of set notation. A.SSE 2) Anticipatory set: Students will discuss what a set is? Procedure: Through example and guided practice, students will complete exercises using set notation. Students will define various symbols and types of sets. Closure: graded worksheet Resources: Slides, textbook, worksheet Adaptations: Work with individual students or small groups as needed, provide shell notes, additional challenge problems Homework: Worksheet 3) Assessment: Informal during group and individual work. Select students to share and justify their solutions and explain strategies for various problems

3. Discrete Unit 1 Lesson Plan 1) SWDAT use Venn Diagrams in mathematical concepts. Standards for mathematical practice 1 2) Anticipatory set: Students will discuss where they have used Venn Diagram. Procedure: Through example and guided practice, students will complete exercises using Venn Diagrams. Students will expand their knowledge from 2 circle to 3 circle diagrams. Closure: Group WS Resources: Slides, textbook, worksheet Adaptations: Work with individual students or small groups as needed, provide shell notes, additional challenge problems Homework: Worksheet 3) Assessment: Informal during group and individual work. Select students to share and justify their solutions and explain strategies for various problems

4. Discrete Unit 1 Lesson Plan  1) SWDAT apply Direct Reasoning (modus ponens), Indirect Reasoning (modus tollens), and Transitive Reasoning (syllogism )  CC.9-12.G.CO.9, CC.9-12.G.CO.10, and CC.9-12.G.CO.11 2) Anticipatory set: Logic video clip and logic puzzle. Procedure: As a class we will learn about different types of logical thinking and analysis. We will complete various examples and exercises utilizing our new ways of thought. Closure: Recap types of thinking. Resources: Slides, textbook, worksheet Adaptations: Work with individual students or small groups as needed, provide shell notes, additional challenge problems Homework: Worksheet 3) Assessment: Informal during group and individual work. Select students to share and justify their solutions and explain strategies for various problems

5. DISCRETE MATHEMATICS UNIT 1 Set Theory & Its Applications 1)Set Notation 2) Venn Diagrams 3) Logic

6. 1) SET NOTATION What is a SET? A SET is a collection of distinct objects called “elements” or “members”. How do we describe a set of elements? M = {a, b, c, d} N = {d, e, f} This means that set M consists of elements a, b, c and d and that set N consists of elements d, e and f. By distinct we mean an element can not be repeated and must be well-defined. For example P = {a, b, b, c} is not a set since element b is not distinct. In addition Q = {the three largest integers} is not a set since “the three largest integers” is not defined.

7. 1) SET NOTATION Sets can be finite (countable) like sets M and N or infinite like the set of even numbers. Sets can also have no elements. How do we show a set with no elements? { } or  NULL SET OR EMPTY SET

8. 1) SET NOTATION M = {a, b, c, d} N = {d, e, f} How do we indicate that an element is in a set? a  M This means that “a” is an element of set M. How do we indicate that an element is not in a set? a  N This means that “a” is not an element of set N.

9. 1) SET NOTATION Sets in mathematics are often described in words or by using mathematical notation. For example, {x  R | x > 2} This means, “x is an element of the real numbers such that x is greater than 2”. We will not be using notation like this in this class!

10. 1) SET NOTATION How do we indicate that all of the elements of one set are contained in another? To show that one set is a SUBSET of another we use the  symbol. For example, suppose set P = {a, b} and set M = {a, b, c, d}, we can say that P  M (P is a subset of M).

11. 1) SET NOTATION M = {a, b, c, d} N = {d, e, f} What does it mean to take the UNION of two sets? The symbol for UNION is  or . M  N means all elements in set M OR set N. So, M  N = {a, b, c, d, e, f}. Please note that element “d” is not repeated.

12. 1) SET NOTATION What does it mean to take the INTERSECTION of two sets? The symbol for INTERSECTION is  or . Suppose M = {a, b, c, d} P = {a, b}. M  P means all elements in set M AND set P. So, M  P = {a, b}.

13. 1) SET NOTATION What does ~ R mean? This means “not” R or all elements in the UNIVERSAL SET not in set R. Suppose that the universal set is all letters in the alphabet and set R is the set of all consonants, including “Y”. ~ R would be the set of vowels, not including “Y” = { A, E, I, O, U}. This is also called the COMPLEMENT of a set. *~ R can also be written R c or.

14. 1) SET NOTATION M = {a, b, c, d} Q = {b, e } What does M - Q mean? The difference between set M and set Q is the elements in set M except for anything in Q that overlaps with M. So, M - Q = {a, c, d}. *Note that we don’t include “-e” since e is not in M.

15. 1) SET NOTATION SYMBOLS: { } Set brackets  Element of a set  Not an element of a set  Null or empty set  Subset of a set  Not a subset of a set  Union of two or more sets  Intersection of two or more sets ˜ Complement or “not” …

16. 1) SET NOTATION EXERCISES : The Universal Set is the digits 0 to 9. M = {1, 2, 3, 4, 5}, N = {0, 2, 4, 6, 8}, P = {1, 3, 7, 9}, Q = {1, 2, 3} 1. M  N = 2. M  N = 3. M  P = 4. ˜ N = 5. N  ˜P = 6. (P  Q) - Q = 7. Is Q  M? 8. Is 3  P?

17. SET “the game” Website: http://www.setgame.com/ Tutorial: http://www.setgame.com/set/index.html The Rules: http://www.setgame.com/set_puzzle_book/index.htm Daily Puzzle: http://www.setgame.com/set/puzzle_frame.htm

18. Complete pg 10-12 in notes

19. Turn to pg 13 in notes

20. 2) VENN DIAGRAMS Let’s say that the universal set is the numbers 1, 2, 3 and 4. A = {1, 2} and B = {2, 3} UNION A  B or A  B means “A union B”, or everything in either set. Answer: {1, 2, 3}

21. 2) VENN DIAGRAMS INTERSECTION A  B or A  B, which means “A intersect B” or only things in both sets. Answer: {2}

22. 2) VENN DIAGRAMS COMPLEMENT ˜A which means “the complement of A” or not A, is everything in the universe outside of A. Answer: {3, 4}

23. 2) VENN DIAGRAMS The Difference between A and B A - B is everything in A except for anything A and B share in common. Answer: {1}

24. 2) VENN DIAGRAMS ˜(A  B) “Not (A union B)” is everything outside of A and B. Answer: {4}

25. 2) VENN DIAGRAMS ˜(A  B) “Not (A intersect B)” is everything outside of the overlap of A and B. Answer: {1, 3, 4}

26. 2) VENN DIAGRAMS How can we show that one set is a subset of another using a Venn Diagram? (C  D) This is equivalent to the conditional statement “If C, then D”. C D

27. 2) VENN DIAGRAMS Shade A  C

28. 2) VENN DIAGRAMS Shade A  (B - C)

29. 2) VENN DIAGRAMS Shade ˜A  B

30. 2) VENN DIAGRAMS Show each of the following situations using Venn Diagrams. 1) A  B.2) A  B =  A B A B

31. 2) VENN DIAGRAMS Practice Problems: http://nlvm.usu.edu/en/nav/frames_asid_153_g_4_t_ 1.htmlhttp://nlvm.usu.edu/en/nav/frames_asid_153_g_4_t_ 1.html?open=instructions&from=category_g_4_t_1.h tml

32. Pg 14 in notes

33. 2) VENN DIAGRAMS

34. 2) VENN DIAGRAMS Pg 15 in notes

35. 2) VENN DIAGRAMS Pg 5 Using a system of equations

36. 2) VENN DIAGRAMS Using a system of equations

37. 2) VENN DIAGRAMS Fill in the Venn Diagram that would represent this data. 150 people at a concert - * 18 people could play none of these instruments. * 10 people could play all three of these instruments. * 77 people could play drums or guitar but could not play piano. * 73 people could play guitar. * 49 people could play at least two of these instruments. * 13 people could play piano and guitar but could not play drums. * 21 people could play piano and drums.

38. 2) VENN DIAGRAMS Fill in the Venn Diagram that would represent this data. 150 people at a concert - * 18 people could play none of these instruments. * 10 people could play all three of these instruments. * 77 people could play drums or guitar but could not play piano. * 73 people could play guitar. * 49 people could play at least two of these instruments. * 13 people could play piano and guitar but could not play drums. * 21 people could play piano and drums.

39. 2) VENN DIAGRAMS pg. 16 in notes A study was made of 1000 rivers to determine what pollutants were in them. * 177 rivers were clean * 101 rivers were polluted only with crude oil * 439 rivers were polluted with phosphates. * 72 rivers were polluted with sulfur compound and crude oil, but not with phosphates. * 289 rivers were polluted with phosphates, but not with crude oil. * 463 rivers were polluted with sulfur compounds. * 137 rivers were polluted with only phosphates. Fill in the Venn Diagram that would represent this data. (Hint: Use a system of equations.)

40. Pg 17 in notes

41. 3) LOGIC Labyrinth Riddle - http://www.youtube.com/watch?v=2dgmgub8mHw

42. 3) LOGIC  Direct Reasoning (modus ponens)  Indirect Reasoning (modus tollens)  Transitive Reasoning (syllogism )

43. 3) LOGIC Symbolic Logic - p is used to represent a premise or hypothesis q is used to represent a conclusion p  q means “if p is true then q must be true”  p means “p is not true”  q means “q is not true” p  q mean “p is true if and only if q is true”  means union or “or”  means intersection or “and”  means therefore

44. 3) LOGIC  Direct Reasoning - If A  B and x  A, then x  B. A B x Example: If you take Discrete Math (set A), then you are smart (set B). Joe (x) takes Discrete Math. What can we conclude? Joe is smart! A  B

45. 3) LOGIC  Direct Reasoning - If A  B and x  B, we cannot be certain that x  A? A B x Example: If you take Discrete Math (set A), then you are smart (set B). Joe (x) is smart. Can we conclude anything else about Joe? NO! Joe could be inside set A or outside set A, but inside set B. A  B x

46. 3) LOGIC  Indirect Reasoning - If A  B and x  B, then x  A. A B x Example: If you take Discrete Math (set A), then you are smart (set B). Jane (x) is not smart. What can we conclude? Jane is not in Discrete Math. A  B ~B  ~A (contrapositive)

47. 3) LOGIC  Indirect Reasoning - If A  B and x  A, it is possible that x  B or x  B. A B xWhat if we know that Jane is not in Discrete Math (set A) - can we conclude that she is not smart (not in set B)? No, we can not conclude this because Jane could be represented by either point x. x A  B  A   B? NO! (inverse)

48. 3) LOGIC  Transitive Reasoning - If A  B and B  C, then we can conclude that A  C. Any element of A is also an element of B and of C. A B C x Example: If you are in Discrete Math, then you are smart. If you are smart then you love math. Joe is in Discrete Math. What can we conclude? Joe is smart and he loves math!

49. 3) LOGIC  Transitive Reasoning - If A  B and B  C, then we can conclude that A  C. Any element of A is also an element of B and of C. A B C x A  B B  C  A  C x  A, x  B, x  C This is also know as a syllogism.

50. 3) LOGIC A BC Does transitive reasoning apply to intersecting sets? If A  B and B  C, does A  C? Not necessarily!

51. 3) LOGIC Use Indirect Reasoning to Solve this Riddle I have three hats, two red and one white. I put one hat on your head and one on your friend’s head, while you both have your eyes closed. When you open your eyes you see a red hat on your friend’s head, and when your friend opens his eyes, he looks at you and is not sure what color his hat is. What color is your hat?

52. 3) LOGIC notes pg 20 SYLLOGISMS AND LEWIS CARROLL Lewis Carroll was a professor of mathematics at Oxford University in England. He studied logic as a vocation, and he played with logic in his writings. His stories of little girls and strange creatures are filled with bad puns and other plays with words, absurd implications, contradictions, and numerous and various offenses to common sense. It is as though he were writing his silly stories as much to amuse himself as to entertain his audiences.

53. 3) LOGIC SYLLOGISMS AND LEWIS CARROLL As a teacher of logic and a lover of nonsense, Carroll designed entertaining puzzles to train people in systematic reasoning. In these puzzles he strings together a list of implications, purposefully inane so that the reader is not influenced by any preconceived opinions. The job of the reader is to use all the listed implications to arrive at an inescapable conclusion.

54. 3) LOGIC SYLLOGISMS AND LEWIS CARROLL Here is one of Lewis Carroll's simpler puzzles. (a) All babies are illogical. (b) Nobody is despised who can manage a crocodile. (c) Illogical persons are despised. What can we conclude? (Hint: Use Venn diagrams to analyze this syllogism.)

55. 3) LOGIC SYLLOGISMS AND LEWIS CARROLL Babies Illogical people Despised people People who can manage A crocodile

56. 3) LOGIC Cogito ergo sum! I think, therefore I am. (René Descarte, 1637 - originally in French - Je pense donc je suis) think exist

57. 3) LOGIC I am not, therefore I think not. (If I do not exist, then I do not think.) (contrapostive) x x Is it possible to exist without the ability to think? Where would that fall on our Venn diagram? What does that mean from a medical standpoint? think exist

58. Notes Pg. 18-19 Reference sheet Pg. 21-22 classwork