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By A. T. Berztiss and Werner Rheinboldt (Auth.)

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Proof. The given system is precisely the Boolean algebra ©, *, ', 0, 1> of Section le, with set B having just two elements, the distinct elements 0 and 1. To prove that the system is a Boolean algebra it suffices to show that {0, 1} is closed under ©, *, and '. 12 (all of Section le) we have the following definitions of operations ©, *, ' for members of {0, 1}: 0 O ©© 1I O © O = O, 0*0 = 0 * 1 == = O' = 1 © I<3 = 1 * 0 = 1, 0 = 0, 0, 1 0 1 = 1; 1 * 1 = 1; R = O. Operations are functions ( © : {0, l } -+ {0, 1} is the function {<0, 0, 0>, <0, 1, 1>, <1, 0, 1>, <1, 1, 1>}), and the three functions are all into (and onto) the set {0, 1}.

Hence the probability of receiving a particular poker hand is 1/2,598,960, and that of receiving a particular bridge hand 1*57 x 1 0 " . 3. The score for 10 throws of a die, with order disregarded, is a 10-selection of a set of six elements. Hence the total number of unordered scores is C(15, 10) = 3003. Here we cannot assume that all selections have an equal probability of occurrence; the events with equal probability are the ordered samples. There is only one sample that results in score 6666666666, but 10 samples result in score 5666666666.

2 m m 1 Z 3 m SUBROUTINE NEXTS (SAMPLE, M) INTEGER SAMPLE(M) K = M I F ( S A M P L E ( K ) . E Q . 4 lh. , it finds all orderings of this selection. If an m-selection of {0, 1} contains n zeros, there are C(m,n ) such samples. Prior to the first call to NEWSAM array SAM must be set to <0, 0, . . , 0>. The m-selection is specified by NZ, the value of which determines the number of zeros in the selection. 3 0 0 samples 0 0 0 1 1 , 0 0 1 0 1 , 0 0 1 1 0 , 0 1 0 0 1 , 0 1 0 1 0 , 0 1 1 0 0 , 1 0 0 0 1 , 10010, 10100, 11000.

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