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Finite Maths Problems
 

Finite Maths Problems

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    Finite Maths Problems Finite Maths Problems Presentation Transcript

    • YOUR FIRST ASSIGNMENT! 1.) Log onto the M118 Oncourse Webpage 2.) Download and read the Class Syllabus. 3.) Download and read the Departmental Course Outline. 4.) Download and save the Powerpoint slides for Week 1. (located on the RESOURCES Page of Oncourse) 5.) Do the Welcome Set on the Webwork Homework System
    • WEBWORK AND BROWSERS!! PLEASE use FIREFOX for doing the Webwork homework. FIREFOX is downloadable for free at firefox.com. Webwork does NOT work correctly when using SAFARI or INTERNET EXPLORER, so please do not use these or you will run into trouble! ALSO, claiming that your computer isn’t working properly is not a valid excuse for not completing a homework assignment. If your computer is not working, then you MUST go to one of the many public computing sites on campus to complete the work.
    • PROBABILITY MODELS (the application of Set Theory to the study of random events)
    • Part 1 – Review of Set Theory DEFN: A SET is a collection of “things”, called elements. TWO WAYS TO DESCRIBE SETS: Method 1: List the elements: NOTE: 3  S but 12  S “” means “is an element of” Method 2: Use a rule: S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} or S = {1, 3, 4, 2, 5, 7, 10, 8, 6, 9} S = {x : x is a positive integer less than 11} read “the set of all things x such that x…”
    • S = { } = the empty set ()  is the smallest set we will need to work with……
    • Q: What is the LARGEST set we will need to work with?
    • The largest set we will need to think about is….. DEFN: A Universal Set is the largest set that needs to be considered for a given “selection problem”.
    • A selection problem: Suppose the members of AC/DC get tired of their name, so they decide to pick some new letters to be their new name. What is a reasonable Universal Set for this problem?
    • How’s about………… U = {a, b, c, … , x, y, z}
    • SIX WAYS to form new sets from other sets:
    • 1.) form a SUBSET of elements from a given set: DEFN: A set A is a subset of a set B if every element in A is also in B. A  B is read “A is a subset of B”, and B  A is ALSO read “A is a subset of B”
    • Example: Let A = { a, b, c, d, e, f, g, h, i, j } B = { c, e, f }, and C = {b, c, e, f, k } Which of the following are true statements? a.) A  B b.) C  B c.) A  A d.)   B
    • Example: Let A = { a, b, c, d, e, f, g, h, i, j } B = { c, e, f }, and C = {b, c, e, f, k } Which of the following are true statements? a.) A  B (False) b.) C  B (True) c.) A  A (true for any A) d.)   B (true for any B)
    • 2.) Form the UNION of two sets A  B = the set of all of the elements  that are either in A or in B “union” or in both. KEYWORD “OR”
    • EX: Let A = { a, b, c, x, y, z } & B = { d, e, f, x, y, z } Then A  B = ??
    • EX: Let A = { a, b, c, x, y, z } & B = { d, e, f, x, y, z } Then A  B = { a, b, c, d, e, f, x, y, z }  NOTE!! We never list elements in a set twice!!!
    • 3.) Form The INTERSECTION of 2 sets A  B = the set of ONLY the elements  that are both in A and in B intersection at the same time! KEYWORD “AND” intersection = “common elements”
    • EX: Let A = { a, b, c, x, y, z } & B = { d, e, f, x, y, z } Then A  B = ??
    • EX: Let A = { a, b, c, x, y, z } & B = { d, e, f, x, y, z } Then A  B = { x, y, z }
    • IMPORTANT WARNING! Given two sets A and B, A.) the set containing the things that are in “A and B” is the intersection of A and B, but B.) the set containing “the things in A and the things in B” is the union of A and B!
    • EX: Let A = { a, b, c } Then A  B = ?? & B = { d, e, f }
    • EX: Let A = { a, b, c } & B = { d, e, f } Then AB=  DEFN: If A  B = , A and B are said to be Mutually Exclusive or Disjoint.
    • 4.) Form a PARTITION of a set: DEFN: A PARTITION of a set X is a collection of mutually exclusive subsets of X whose union is the entire set X. English Translation: Partitioning a set X means chopping the set up into non-overlapping pieces such that when you put them back together, you get X again.
    • EG) Given X = {a, b, c, d, e, f, g, h }, Which of the following form a partition of X ? #1 A = { a, c, d} B = {b, e, h} C={g} #2 A = { a, c, d} B = {b, e, g, h} C = { f, g } #3 A = { a, c, d} B = {b, e, h} C = { f, g }
    • 5.) Form the COMPLEMENT of a subset of a universal set U. Given A  U, then A’ =  set of those elements in U that are NOT in the set A. complement KEYWORD “NOT”
    • EX: Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } and A = {2, 3, 4, 5, 7, 9 } Then A’ = ??
    • EX: Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } and A = {2, 3, 4, 5, 7, 9 } Then A’ = { 1, 6, 8, 10 }
    • 6.) Form the CARTESIAN PRODUCT of two sets A and B. DEFN: The Cartesian Product of two sets, A, and B, is the set of all ordered pairs of elements, (a, b), where a  A and b  B, or A x B = { (a,b): a  A and b  B }
    • Example: Techston McNorgly finally gets up the nerve to ask out that girl from his Chemistry class…and for some unknown reason…..she says “Yes”!
    • Techston decides his date will be dinner then a movie… D = { Taco Bell, Burger King, White Castles } M = { Deuce Bigalow IV, Gigli 2 } Q: What are all the possible DATES ??
    • Techston decides his date will be dinner then a movie… D = { Taco Bell, Burger King, White Castles } M = { Deuce Bigalow IV, Gigli 2 } Q: What are all the possible DATES ?? A: a “DATE” = (d,m), where d  D and m  M, So… “the set of all possible dates” = D x M Q: How do we find D x M ?
    • An “OPTION” or “OUTCOME” TREE TB START BK WC Pick a dinner option,
    • An “OPTION” or “OUTCOME” TREE TB START BK WC DB { (TB, DB), G (TB, G), DB (BK, DB), G (BK, G), DB (WC, DB), G (WC, G) } =DxM Pick a dinner option, then pick a movie option
    • EX) Let U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }, with subsets A = { 2, 4, 5, 6, 8 } B = { 1, 3, 8, 10 } and C = { 5, 9 } List the elements in the following sets: 1) C x B 2) A ’  (B  C)’
    • QUICK REVIEW Basic Concepts: 6 ways to form new sets a) set b) element c) empty set d) universal set 1) subset 2) union 3) intersection 4) partition 5) complement 6) cartesian product
    • VENN DIAGRAMS & COUNTING ELEMENTS IN SETS
    • Suppose a Universal Set U has two subsets, A and B. We can represent the relationships between these various sets visually using a VENN DIAGRAM: whole rectangle = universal set U A B One can think of a Venn diagram as an “element storage bin”. (you know, a place to store your elements)
    • Suppose a Universal Set U has two subsets, A and B. We can represent the relationships between these various sets visually using a VENN DIAGRAM: whole rectangle = universal set U the set A is indicated by the yellow region A A B
    • Suppose a Universal Set U has two subsets, A and B. We can represent the relationships between these various sets visually using a VENN DIAGRAM: whole rectangle = universal set U the set A’ is indicated by the yellow region A A B
    • Suppose a Universal Set U has two subsets, A and B. We can represent the relationships between these various sets visually using a VENN DIAGRAM: whole rectangle = universal set U the set A’ is indicated by the yellow region A A B Q: What about the set A  B’ ?
    • How to shade  and  of ANY two sets in Venn Diagrams: 1) Shade each set using a different color/pattern. 2) To shade the union (), shade everything that is already shaded in one pattern OR the other. 3) To shade the intersection (), shade only the portion of the diagram that is already shaded using both colors/patterns together.
    • EX) Shade each of the following sets in a Venn Diagram: 1.) A  B 2.) A  B 3.) A’  B 4.) A’  B 5.) (A  B) ’ 6.) A ’  B ’
    • EX) Shade each of the following sets in a Venn Diagram: 1.) A  B 2.) A  B 3.) A’  B 4.) A’  B (STUDY HINT: It’s a good idea to memorize the four disjoint parts of a 2 set Venn Diagram.) 5.) (A  B) ’ 6.) A ’  B ’ DeMORGAN’S LAWS: 1) 2) (A  B)’ = A’  B’ (A  B)’ = A’  B’
    • PROBLEM: SHADE the sets (A  B’)  C and (A  B’)  C A B C
    • PROBLEM: SHADE the sets (A  B’)  C and (A  B’)  C SOLUTION: A B C RED = A  B’
    • PROBLEM: SHADE the sets (A  B’)  C and (A  B’)  C SOLUTION: A B RED = A  B’ WHITE = C C
    • PROBLEM: SHADE the sets (A  B’)  C and (A  B’)  C SOLUTION: A B YELLOW = (A  B’)  C C
    • PROBLEM: SHADE the sets (A  B’)  C and (A  B’)  C SOLUTION: A B RED = A  B’ WHITE = C C
    • PROBLEM: SHADE the sets (A  B’)  C and (A  B’)  C STUDY HINT: It’s a good idea to memorize the eight disjoint parts of a 3 set Venn Diagram. SOLUTION: A B BLACK = (A  B’)  C C C
    • Three Counting Rules for sets: terminology: Given a set S, n(S) = number of elements in S Example: Suppose X = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } is partitioned into three subsets A, B, and C as follows: A = { 1, 3, 5 } B = { 2, 4, 7, 8, 9 } C = { 6, 10 } HOW IS n(X) related to n(A), n(B), and n(C) ??
    • RULE 1) THE PARTITION SUM RULE: If a set X is partitioned into subsets, then the number of elements in X can be found by adding together the numbers of elements in each of the subsets of the partition. A Special Case of the Partition Sum Rule: If A  U, then n(A) + n(A’) = n(U) or n(A’) = n(U) - n(A)
    • Example: Suppose the universal set U = { a, b, c, d, e} has two subsets, A = { a, c, d } and B = { b, c, d, e} Q: How are n(A), n(B), n(A  B), and n(A  B) related??? HINT: Use a Venn Diagram to investigate!!!
    • RULE 2) THE UNION/INTERSECTION RULE: Given any two sets A and B, it is always true that n(A  B) = n(A) + n(B) - n(A  B) and n(A  B) = n(A) + n(B) - n(A  B)
    • EX: Recall Techston’s big date…how are n(D), n(M), and n(DxM) related? TB START BK WC DB { (TB, DB), G (TB, G), DB (BK, DB), G (BK, G), DB (WC, DB), G (WC, G) } Pick a dinner option, then pick a movie… =DxM
    • RULE 3: CARTESIAN PRODUCT RULE: The number of elements in the cartesian product of two sets A and B is the product of the number of elements in each set, or n(A x B) = n(A)●n(B). TB START BK WC DB { (TB, DB), G (TB, G), DB (BK, DB), G (BK, G), DB (WC, DB), G (WC, G) } Pick a dinner option, then pick a movie… =DxM
    • RULE 3: CARTESIAN PRODUCT RULE: The number of elements in the cartesian product of two sets A and B is the product of the number of elements in each set, or n(A x B) = n(A)●n(B). Does this idea generalize? Suppose the date goes so well that Techston and his date decide to spend more time together after the movie….
    • TB START BK WC DB G DB G CT WM { (TB, DB, CT), (TB, DB, WM), CT WM CT … WM CT DB WM G WM CT CT WM (WC, G, WM) } = S Pick dinner option, then pick movie, then pick after-movie activity n(D x M x A) = n(D)·n(M)·n(A)
    • Counting Rules: QUICK REVIEW 1) Partition Sum Rule 2) Union/Intersection Rule 3) Cartesian Product Rule The Rule of Thumb: When one of the 3 rules doesn’t seem to be working, try making a Venn Diagram for the situation!
    • EX1) Let A and B be subsets of U, with n(U) = 40, n(A) = 16, n(B) = 27, and n(A  B) = 9. Find… a) n(A  B) b) n(A’) c) n(B’  A)
    • EX2) Let A and B be subsets of U, with n(U) = 30, n(A) = 14, n(B) = 18, and n(A’  B’) = 10. Find… a) n(B’) b) n(A  B) c) n(A  B’ )
    • EX3) There are 270 people enrolled in class: 140 are in the wrong room, 180 would rather be sleeping, 100 are in the wrong room AND would rather be sleeping. a) How many people are in the correct room? b) How many people are in the wrong room, but would NOT rather be sleeping?
    • EX4) A universal set U is partitioned into four subsets, A, B, C, and D. If n(U) = 75, n(A) = n(B), n(B) = 3•n(C) and n(C) = 2•n(D), what is n(C) ? HINT: Use a variable to represent the number of elements in the SMALLEST set!!
    • EX5) A universal set U is partitioned into two subsets, X and Y. If X = E x F x G, and Y = G x E, With n(E) = 3, n(F) = 2, and n(G) = 4, what is n(U)?
    • EX6) Let U be a universal set with subsets A, B, and C. If n(U) = 80, n(A) = 26, n(B) = 34, n(C) = 30, n(A  B) = 10, n(A  C) = 8 n(B  C) = 12, and n(A  B  C) = 3 Find the following: a) n(A’  B’  C’) b) n(A  B) c) n(A’  B  C) d) n(A’  B  C’)
    • EX7) The 270 students in a finite math class are surveyed. Of these, 215 feel their 195 feel their 220 feel their 180 feel their 195 feel their 160 feel their 25 people feel instructor is a Geek (G) instructor is a Dweeb (D) instructor is a Nerd (N) instructor is both a Geek and a Dweeb instructor is both a Geek and a Nerd instructor is a Geek, a Dweeb, and a Nerd their instructor is neither a Geek, Dweeb, nor Nerd How many of these 270 students… a) feel their instructor is both a Dweeb and a Nerd, but not a Geek? b) feel their instructor is a Geek but neither a Dweeb nor a Nerd? c) feel their instructor is either a Dweeb or a Nerd, but not a Geek?