Assume the Boolean matrix below is MR, and that MR represents the relation R where R represents the connecting flights that an airline have between four cities: a,b,c,d. The 1 in row a column b means there is a flight from city a to city b. In general, there is a 1 in row x, column y if and only if there is a connecting flight between city x and city y. That is, the rows of the matrix represent the cities of the origins of the flights and the columns represent the destination cities.

a b c d
Let MR = 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 Manchester, let b stand for the airport in Boston, let c stand for the airport in Chicago, let d stand for the airport in Denver. Is there a flight from Denver to Chicago?

(ii) Compute and MR^2 and MR^3 (use Boolean arithmetic). What do these Boolean products give you? In other words, what do the Boolean entries in the matrices MR^2 and MR^3 mean?

(iii) Now call the given Matrix A and compute A^2 and A^3 using regular, not Boolean, arithmetic.

(iv) What does MR + MR^2 + MR^3 + MR^4 give me?