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Ak ) of A is viewed as a preferential ordering of the alternatives in A. All things in A1 are tied for first place, all things in A2 are tied for second place, etc. Economists call the blocks Ai “indifference classes,” since a decisionmaker is indifferent between the alternatives comprising a given block Ai . ) Here is a partial table of values of Pn : n: 0 1 2 3 4 5 Pn : 1 1 3 13 75 541 Although it is hard to guess from this table, or even from an extended version of this table, the numbers Pn are generated by a fairly simple recursive formula.

If a red die and a clear die are tossed, in how many ways may the sum 9 be attained? Elementary probability texts usually solve this problem by brute force, deriving the answer 4. Here is another approach, which not only solves the problem for the sum 9, but for all possible sums. Simply multiply the polynomials (x + x2 + x3 + x4 + x5 + x6 )(x + x2 + x3 + x4 + x5 + x6 ) = x2 + 2x3 + 3x4 + 4x5 + 5x6 + 6x7 + 5x8 + 4x9 + 3x10 + 2x11 + x12 . Note that the coefficient of x9 is 4, the answer to our problem.

Xn } consists of a set of points in the plane, arbitrarily labeled x1 , . . , xn , with an arrow from xi to xj whenever xi ρxj and a “loop” at xi whenever xi ρxi . 1: Binary Relations (ii) The matrix representation of the binary relation ρ on S = {x1 , . . , xn } is the n × n matrix M = (mij ) where mij = 1 0 47 if xi ρxj if ∼ xi ρxj CHAPTER 8. BINARY RELATIONS Here are the matrix representations of the aforementioned three binary relations:       0 0 0 1 0 0 1 1 1 0 0 0  0 1 0   1 1 1 .

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