By D. G. H. B. Lloyd and C. Plumpton (Auth.)
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Truth value of y 0 Truth value of 1 Truth values of x v y 0 0 1 1 1 1 Now looking in the same way at the other connective conjunction ("and"), which makes a true compound statement only if both the simple statements are true. We may write out the truth table in the two different ways as follows: X y 0 0 0 l 1 0 1 l x A 0 0 0 1 y y A X 0 1 0 0 0 0 1 1 The Boolean nature of the mathematics involved here should now be apparent. We have a two-element set, the truth values 0 and 1, and we have two operations—the connectives v and A whose tables of definition 56 M O D E R N SYLLABUS ALGEBRA are precisely those encountered in the two-element Boolean algebra under Section 3.
The Double inverse law VXGJB (A'" 1 )" 1 = x. 51 BOOLEAN ALGEBRA B10 De Morgan's laws Vx,yeB (a) (x v y)'1 = x'1 A y'1 ( b ) M j ) - 1 =x-iv y~K Some of these laws are familiar to the reader because their set theory versions have already been encountered in Chapter 1. In that chapter we were able to show that starting from the definitions of union and intersection it is possible to prove the validity of the various set theory laws. Similarly in the more general case of any Boolean algebra it is possible, from the basis of the four axioms, to prove any of these other results.
If we tried to produce a diagram illustrating the set when the domain is the set of real numbers a difficulty at once becomes apparent; the resulting diagram is identical with that for the domain of rationals because we cannot distinguish between points placed close together and a continuous line. These aspects of the differences between different types of number will be considered in same detail in Chapter 6. For the moment we merely note that this diagrammatic illustration of solution sets, whilst useful, must be treated with care as it has serious limitations.