# A Boolean Algebra: Abstract and Concrete - download pdf or read online

By A.P. Bowran

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5/2 Exercise. 5/3 and and also has its dangers. , an empty set. 9/1 Exercise. Prove that: (i) If A £ B for all A, then B = 1 (ii) .............. B, then A = 0 (iii) ....... , then A. B = A. 28? Prove it. 13/1 Exercise. From the statements: (i) All racing motorists are quick-witted. (ii) Plato was a profound thinker. 3 58 A BOOLEAN ALGEBRA (iii) All philosophers are profound thinkers. (iv) Nobody is both quick-witted, and also a profound thinker. can it be deduced that: (a) Plato was a philosopher (b) , , not a racing motorist?

We now extend this to methods of finding n(F)/n(P) by establishing a one-to-one relation between the members ofF, F 1 ; P, P 1 , when the number of members of the sets is not finite. A c X B Fig. (i) A line AB is divided at random by a point X into two parts (Fig. (i)). What is the probability (p) that the part AX shall be less than the part XB? - More generally, the probability that a random point on AB, Fig. (ii), shall lie between X and Y is length of XY length of AB A X y B Fig. (ii) and the probability that a random point inside R, Fig.

2 We now consider an indefinite number of 'unknown' or 'variable' elements AI> A2 , A3 , ••• in a Boolean algebra which can take either of the values 0 or 1, and the corresponding values-also 0 or 1-of any given function of these variables. Exercise. •. take the value 0 or 1, then every function of these variables will take one of these values. 3 One application of such an algebra is in symbolic logic. Here the elements are 'truth values' of statements. If p represents a statement, then 'p is true' is represented by P = 1, and 'p is false' by P = 0.