Note on Carnap's Relational Asymptotic Relative FrequenciesAuthor(s): Frank HararySource: The Journal of Symbolic Logic, Vol. 23, No. 3 (Sep., 1958), pp. 257-260Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2964284 .

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THE JOURNAL OF SYMBOLIC LOGIC Volume 23, Number 3, Sept. 1958

NOTE ON CARNAP'S RELATIONAL ASYMPTOTIC RELATIVE FREQUENCIES'

FRANK HARARY

A (binary) relation is a collection of ordered couples. Two relations are isomorphic if there is a 1-1 correspondence between their fields which preserves the ordered couples. Tsomorphism between relations is itself an equivalence relation, and a structure is an isomorphism class of binary relations.

Carnap ([1], p. 124) asks certain questions concerning (both the exact and) the asymptotic value of the relative frequency that a relation on p objects satisfies certain properties. Among the most interesting special cases are the asymptotic value of the relative frequency that a binary relation on p objects be (a) symmetric, (b) reflexive, (c) transitive irreflexive anti- symmetric, and (d) symmetric irreflexive. These results already appear either in, or almost in, the literature, in various disguises. The object of this expository note is to bring them to light, especially since some of the references may not be generally known to logicians.

Carnap ([1], p. 124) points out explicitly the correspondence between binary relations and linear graphs. Davis [2] and Harary [4] obtains precise results for the number of structures with certain properties using the language of relations and graphs respectively; but these do not supply the asymptotic values directly. The asymptotic results which are most useful here are either contained in, or are straight-forward extensions of, the formulas presented in Ford and Uhlenbeck [3], which in turn are based on unpublished work of G. Polya. In addition, we utilize a result of L. Moser on transitive relations, which appears in Wine and Freund [5].

Notation Let the number of structures on p objects satisfying the respectively

indicated properties be denoted as follows:

Number Kind of structures

R(p) all R'(p) irreflexive s(p) symmetric s'(p) symmetric irreflexive r(p) reflexive t(p) transitive irreflexive antisymmetric

Received June 20, 1958. This work was supported by a grant from the National Science Foundation.

257

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258 FRANK HARARY

Furthermore let R(p, q) be the number of structures on p objects with exactly q ordered couples; with s(p, q), etc. having analogous meanings.

Results from the literature. Asymptotic formulas (1) and (2) appear in [3], p. 164.

(l) ~~~~S'(p, q) I (f(P- 1)/2)

(2) S'(p) 2v(v-1)/2

Formula (1) holds for large p and for 0 < q < p(p- 1)/2. More precisely, Polya (unpublished) proves that (1) is valid if Iq-p(p- 1)/41 < cp where c is a positive constant. The precise range of q for the validity of (1) is not known, but the great majority of relations is included. One obtains (2) at once from (1) by summing over q, noting that q is always even.

It is then easily seen that

(3) S ~ p 2P~l/

(4) r(p) = R'(p) -2v(v-1),

(5) R(p)p 2V2.

The first equality in (4) is contained in Davis [21, Theorem 3. Moser has shown (cf. [5]) that

(6) t(p) = pl(p

As variations in (1), we have (with analogous limitations on the range of q):

(7) (P, ) I! (P 1)/2)

(8) r(p, q) = R'(p, q) I (P(P- 1))

(9) R(p, q) I p(p+ 1)

Combining these results, we obtain the following statements. The asymp- totic relative frequency that a relation on p objects be symmetric is

l 1) E2s(p) 1

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RELATIONAL ASYMPTOTIC RELATIVE FREQUENCIES 259

symmetric irreflexive is

(l l) ~~~~~~~R(p) 2Xv(XV+1)/2;

reflexive (or irreflexive) is

r(p) 1 (12) ~~~~~~R(fi) 2p'

and transitive irreflexive antisymmetric is

(13) _t(p) (p- 1) ! (2p

R(p) 2P(P-1) p

Some analogous ratios for the asymptotic relative frequencies of struc- tures on p objects having exactly q ordered couples can be obtained by combining formulas (1), (7), and (8) with (9).

Further remarks in the language of graph theory. Referring to [4] for preliminary definitions not included here, a connected

graph2 is one in which each pair of points is joined by a path, and a block is a connected graph with no cut points. The group of a graph G is the collection of all automorphisms of G, i.e., one-to-one mappings of the set of points of G onto itself which preserve the lines of G. Trivially, the identity mapping is always an automorphism. If there are no other automorphisms, we call G an identity graph. By definition, graphs with p points and k lines correspond biuniquely with irreflexive structures on p objects having q = 2k ordered couples. It is shown in Ford and Uhlenbeck [3]that asymp- totically, "almost all" graphs are connected, are blocks, and are identity graphs. The translation of these results to the language of relations3 is routine.

REFERENCES

[1] R. CARNAP, Logical foundations of probability, Chicago, 1950. [2] R. L. DAVIS, The number of structures of finite relations, Proceedings of the

American Mathematical Society, vol. 4 (1953) 486-495.

2 A complete graph is one in which every two distinct points are adjacent to each other. A relation is usually called connected if for any two distinct objects in its field, at least one of the two ordered couples lies in the relation. Hence "connected relations" are coordinated to "complete graphs," and not to connected graphs.

3 Corresponding to the terms "relation" and "structure," we use "labeled graph" and "graph" respectively. In these terms, an equivalence relation corresponds to a labeled graph in which every connected component is complete. Hence the number of equivalence structures on p objects is =(p), the number of partitions of the positive integer p into positive summands without regard to order.

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260 FRANK HARARY

[3] G. W. FORD and G. E. UHLENBECK, Combinatorial problems in the theory of graphs, IV Proceedings of the National Academy of Sciences, vol. 43 (1957) 163-167.

[4] F. HARARY, The number of linear, directed, rooted, and connected graphs, Trans- actions of the American Mathematical Society, vol. 78 (1955) 445-463.

[5] R. L. WINE and J. E. FREUND, On the enumeration of decision patterns involving n means, Annals of mathematical statistics, vol. 28 (1957) 256-259.

THE UNIVERSITY OF MICHIGAN AND THE INSTITUTE FOR ADVANCED STUDY

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