5. Write the numbers of the Venn diagram sections that represent the following p

Question

5. Write the numbers of the Venn diagram sections that represent the following parts, where 8 is the whole set and 9 is 0. For example, An B = {2,4} (a) AnBnc (b) A (BuC) (c) A(BnC) (d) AnAc (e) AU(AnB) (f) Try to use Venn diagram to explain why An (BU C) = (A n B) U (An C) 6. (adapted from Ross, 2.5) A system is comprised of 5 components, each of which is either working or failed. Consider an experiment that consists of observing the status of each component, and let the outcome of the experiment be given by the vector (zi, x2, x3, x4, r5), where xi is equal to 1 if component i is working and is equal to 0 if component i is failed (a) Suppose that the system will work if components 1 and 2 are both working, or if compo- nents 3 and 4 are both working, or if components 1, 3, and 5 are all working. Let W be the event that the system will work. Specify all the outcomes in W (b) Let A be the event that components 4 and 5 are both failed. Write out all the outcomes in the event An W

Explanation / Answer

8. a. Ac B C : 6

b. A (B U C) : 1

c. A (B C) : 1,2,3

d. A Ac : 9

e. A U (A B) : 1,2,3,4

f. In the Venn diagram, A (B U C) = 2,3,4

(A B) = 2,4 (A C) = 3,4

=> (A B) U (A C) = 2,3,4

Thus A (B U C) = (A B) U (A C).

6. (a) W = {(1,1,0,0,0), (1,1,0,0,1), (1,1,0,1,0),(1,1,1,0,0),(1,1,0,1,1),(1,1,1,0,1),(1,1,1,1,0),(1,1,1,1,1),(0,0,1,1,0),(0,0,1,1,1),(0,1,1,1,0),(1,0,1,1,0),(0,1,1,1,1),(1,0,1,1,1), (1,0,1,0,1)}

(b) A = {(0,0,0,0,0),(0,0,1,0,0),(0,1,0,0,0),(1,0,0,0,0),(0,1,1,0,0),(1,0,1,0,0),(1,1,0,0,0),(1,1,1,0,0)}

A W = {(1,1,0,0,0),(1,1,1,0,0)}

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