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# Boolean Matrix Questions

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Assume the Boolean matrix below is MR and that MR represents the relation R where R represents the connecting flights that an airline has between 4 cities: a, b, c, and d. so there is a 1 in row x column y iff there is a connecting flight between (from) city x and (to)city y That is, the rows of the matrix represent the cities of the origins of the flight and the columns represent the destination cities.
a b c d
a [1 1 0 0]
b [0 1 1 0]
c [0 0 1 1]
d [1 1 0 0]

(i) Let a stand for the airport in the city of Manchester, let b stand for the airport in Boston, c stand for the Chicago airport, d for the airport in the city of Denver. Is their a flight from Manchester to Chicago?
(ii) Compute and interpret the Boolean products: MR 2, and MR 3. (Remember to use Boolean arithmetic)
(iii) Now call the given matrix A and compute A2 and A3 using regular not Boolean arithmetic. What do these products give you.
(iv) Again call the given matrix A and assume there are 3 flights Boston to Chicago and four from Chicago to Denver and compute A2.

What does MR + MR 2 + MR 3 + MR 4 give you?

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## Boolean product of matrices, One to one function

** Please see the attachment for complete questions (1 and 2) **

1. Find the Boolean Product of matrices A and B.

2. Given matrix A, compute:
(a) A^-1 (A-inverse)
(b) (A^-1)^3

3. Solve the following systems of equations.
x1 + x2 = 0
-x1 + x2 + x3 = -1
-1x2 + x3 = 2

4. (a) Define the function f: R --> R by f(x) = x^3 + 4.
Briefly explain why f is a 1-1 (one-to-one) function. No proof necessary, just an explanation in some detail.

(b) Is the function g: R --> Z defined by g(n) = ceiling(n/2) a one to one function? Explain.

(c) Briefly explain what f^-1 means in general and then find f^-1 for the function f in part a.

5. Expand (A + B)(A - B). Use the procedures of basic matrix laws.

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