1. ENM 503 Block 1 Algebraic Systems Lesson 3 – Modeling with Sets Sets - Why do we care? The Application of Sets – a look ahead 1 Narrator: Charles Ebeling

2. Some Uses of Sets Logic – analyzing complex relationships Optimization – defining feasible regions Modeling random processes Probability theory Reliability 2

3. Let’s solve a problem Twenty-four dogs are in a kennel. Twelve of the dogs are black, six of the dogs have short tails, and fifteen of the dogs have long hair. There is only one dog that is black with a short tail and long hair. Two of the dogs are black with short tails and do not have long hair. Two of the dogs have short tails and long hair but are not black. If all of the dogs in the kennel have at least one of the mentioned characteristics, how many dogs are black with long hair but do not have short tails? 3

4. This solution is for the dogs 9 - x + 2 + 1 + 1 + 2 + x + 12 - x = 24 27 - x = 24 x = 3 4

5. Modeling Solution Sets (optimization) Consider the following sets where x = production level for product A and y = production level for product B A = {(x,y)| x >= 0 }; B = {(x,y)| y >= 0 }; C = {(x,y)| y <= 4 }; D = {(x,y)| x + y <= 5 }; E = {(x,y)| x <= 3 }. Then A  B  C  D  E = x y 4 3 x + y = 5 5

6. Modeling Solution Sets – 2 F = {x,y| x,y are integers} Then A  B  C  D  E  F = {(0,0), (0,1), (0,2), (0,3), (0,4), (1,0), (1,1), (1,2), (1,3), (1,4), (2,0), (2,1), (2,2), (2,3), (3,0), (3,1), (3,2) } 6 x y 4 3 x + y = 5 A = {(x,y)| x >= 0 }; B = {(x,y)| y >= 0 }; C = {(x,y)| y <= 4 }; D = {(x,y)| x + y <= 5 }; E = {(x,y)| x <= 3 };

7. Disjoint Sets A = {(x,y)| x >= 0 }; B = {(x,y)| y >= 0 }; C = {(x,y)| x+ y = 5 } x y 4 x + y = 4 x + y = 5 Feasible region = A  B  (C  D) = A  B   =  7

8. Modeling Random Processes Let S = the sample space, the set of all possible outcomes from a random process (i.e. the Universe) Let E i be an observed outcome (random event) of the random process Then E i is an element of the sample space, and S = {E 1, E 2, …, E n } A = E i  E j and E i  E j are compound events I shall need an example of this. 8

9. Modeling Random Processes Example 1 Let S = a sample space, the outcome from tossing a pair of coins S = {(H,H), (H,T), (T,H), (T,T)} Let E 1 = a random event (outcome), obtaining two heads E 1 = {(H,H)} Let E 2 = a random event (outcome), obtaining at least one head E 2 = {(H,H),(H,T),(T,H)} 9

10. Modeling Random Processes Example 2 Let S = the set of all possible outcomes from selecting three parts from a parts bins containing both defective and non-defective parts. S = {(N,N,N), (N,N,D), …, (D,D,D} What is the size of the sample space? Let E = the event, at least one part is defective E = {(N,N,D), (N,D,N), (D,N,N), (N,D,D), (D,N,D), (D,D,N), (D,D,D)} Let F = the event, exactly two parts are defective F = {(N,D,D), (D,N,D),(D,D,N)} 10

11. Modeling Random Processes Example 3 Random process: select at random a freshman from the UD fall 2006 class to deliver a keynote address Then the sample space S = the set of all freshman Let A = the random event, a male is selected Let B = the random event, a student having a GPA of 3.5 or better selected Let C = the random event, a student from Ohio is selected Then the event D, a male having a GPA below 3.5 and not from Ohio is given by: D = A  B’  C’ Why did they select me? 11

12. M Yet Another Example Consider a random process in which a manufactured product is selected at random. Products coming off the assembly line may or may not meet design specifications. In addition, these products may or may not pass final inspection. Let S = set of all products coming off assembly Let M = the event, product meets specification Let P = the event, product passes inspection. P 12

13. Continuing the Example Then S = (M  P)  (M’  P)  (M  P’)  (M’  P’) ( a partition) where (M  P) = the event, the product meets specs and passes inspection, (M’  P) = the event, the product does not meet specs and passes inspection, (M  P’) = the event, the product meets specs and does not pass inspection, (M’  P’) = the event, the product does not meet specs and does not pass inspection 13

14. Applications in Reliability -1 A system consists of n components and the sample space is the set of all possible states of the components where the components are either operating or failed E i = the event, component i does not fail, and R = the event, the system does not fail where the system is composed of n components If the components are in series (i.e. if any one component fails the system fails, then R = E 1  E 2  n 1 2n … 14

15. Applications in Reliability - 2 E i = the event, component i does not fail, and R = the event, the system does not fail where the system is composed of n components If the components are in parallel (i.e. if any one component does not fail, the system does not fail, then R = E 1  E 2  n = ( E’ 1  E’ 2  ’ n )’ 2 1 n 15

16. More Reliability An electronic assembly consists of 5 components arranged in a series – parallel circuit as shown: A C D B E Define the events A, B, C, D, and E as the events components A, B, C, D, and E do not fail respectively. The assembly will fail only if there is no operating path from left to right through the network. Express the event F, the assembly does not fail, in terms of the events A, B, C, D, and E. F = [(A  B)  (C  D)]  E 16

17. An alternate approach to More Reliability Let S = {A, B, C, D, E}, then define the power set, P S = { , {A}, {B}, {C}, {D}, {E}, {A,B}, {A,C}, {A,D}, {A,E}, {B,C}, {B,D}, {B,E}, {C,D}, (A,B,C}, {A,B,D}, {A,B,E}, {A,C,D}, {B,C,D}, {B,C,E}, {C,D,E}, {A,B,C,D}, {A,B,C,E}, etc. } Then F = {{A,B,C,D,E}, {A,B,E}, {C,D,E}, {B,C,D,E}, {A,C,D,E}, {A,B,D,E}, {A,B,D,E}} A C D B E 2 5 = 32 elements i.e. set enumeration 17

18. The End of Modeling with Sets Next: Fundamentals beginning with the numbers 1,2,3,… 18