By A.P. Bowran
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Extra resources for A Boolean Algebra: Abstract and Concrete
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? Prove your deductions.
3, p. 44 to give outputs S1, C2 , controlled by A1 , B1, C1, is called an adder. 11 (iv) shows us that a series of such adders will enable us to add any two numbers. 16 Exercise. , C 1 = A0 • B0 = 1 and S 0 = A0 Ll B0 = 1 in words, and check their truth. 17 Exercise. Show that the use of two adders and a union element to form an adder is really saying, 'in order to add x, y, and z, first add x andy, and then add z to the answer'. 18 Exercise. 13 Fig. 1 makes no difference. Why? 19 Exercise. 14.
31 We can sec that, provided we can put the members of an infinite set into a definite ordered sequence, we can show that the set is denumerable. That the soa integers, positive or negative, is a denumerable set is shown by the sequence 0, 1, -1, 2, -2, 3, -3, ... and it is, in fact, possible to prove that the soa algebraic numbers is denumerable. 32 This introduction to the concept of sets gives us the first subject of which our algebra is a mathematical model (an abstract algebra whose elements, operations, and postulates fit the subject).