[Article] Prediction and Entropy of Printed English by Claude E. Shannon

By Claude E. Shannon

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W e , however, reserve this notion for sets with composition laws, cf. Chapter 3, and consider order relations a par with any other relations. fflius, in a chain every couple of elements is comparable. 22 Sets In the following we must acquaint ourselves with a simple but somewhat confusing terminology used in connection with posets. Suppose that the poset X has an element / such that f for every x EL X. For obvious reasons, we call / the first element t of X. Similarly, if there exists an element / such that χ ^ / for all x G X, then / is called the last elements of X.

Fflius, in a chain every couple of elements is comparable. 22 Sets In the following we must acquaint ourselves with a simple but somewhat confusing terminology used in connection with posets. Suppose that the poset X has an element / such that f for every x EL X. For obvious reasons, we call / the first element t of X. Similarly, if there exists an element / such that χ ^ / for all x G X, then / is called the last elements of X. From antisymmetry it easily follows that both the first and last element, if it exists, is unique.

4) 1 It is customary to call / the inverse of /. The situation is visualized in Fig. 6. 5d) Γ\Βι-Β2) = Γ(Βι)-Γ(Β2). ] It is important to observe that, whereas / " as defined above is a function from 0>( Y) to ^ ( X ) , it is not a function from Y to X. Indeed, let Β ={b} consist of a single element b G Y. In general, there will be several elements au a2,... G X which are mapped on b. Furthermore, there may exist an element b G Β such that no element x G X is mapped on b. (Then r 1 Fig. 6. ) However, suppose f:X-*Y is bijective.

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