By A.P. Bowran
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Additional resources for A Boolean Algebra: Abstract and Concrete
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. Ambiguity rarely follows from letting P represent the actual statement as well as its truth value. The statement A = B means that these two statements, A and B, have the same truth value, not that they are the same statement.
28 In every case here we find the same difficulty-a set which is equivalent to a true subset of itself, and the 'number of members' of such a set cannot be treated in the usual way. Euclid's axiom 'the whole is greater than any of its parts' does not apply to such numbers. This was taken as the criterion by which to distinguish between 'finite' and 'infinite' numbers. If a set has no proper subset equivalent to itself, it is a finite set, and has a finite number of members. If it has a proper subset equivalent to itself, we say that the number of its members is infinite.
3 hold. 10 Exercise. 11 Exercise. 10 (e) can be written as A = B', and (f) as A= B. 12 Example. The following problem, 'Braintwister No. 138' by D. P. St. Barnard, appeared in The Observer. Fixing The Forecast 'Weather forecasts that cover only the next 24 hours are just not good enough. : If fine today, it will be windy tomorrow. If wet today, it will be fine tomorrow. If today is cold, humidity today will be high. If hot today it will be calm tomorrow. If calm today, it will be hot tomorrow. If windy today, humidity tomorrow will be low, and tomorrow will be wet.
A Boolean Algebra: Abstract and Concrete by A.P. Bowran