THE JOURNAL OF SYMBOLIC LOGIC
Volume 23, Number 3, Sept. 1958
NOTE ON CARNAP'S RELATIONAL ASYMPTOTIC RELATIVE
FREQUENCIES 1
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. Isomorphism 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 antisymmetric, 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on unpublished work of G. Polya.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
all
irreflexive
symmetric
symmetric irreflexive
reflexive
transitive irreflexive antisymmetric
R(P)
R'(P)
s(fi)
s'(p)
r(j>)
t(P)
Received June 20, 1958.
This work was supported by a grant from the National Science Foundation.
1
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.
(2)
s'{p) ~-i-2*<*- 1 >/ 2
p\
Formula (1) holds for large p and for 0 <^q <^.p(p—l)/2. More precisely,
Polya (unpublished) proves that (1) is valid if \q—p(p—1)/4| < 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
s( £)^,_l_2*<*+W2,
(3)
pi
(4)
r(p)=R'(p)~^-2*(>-»,
pi
(5)
R(fi)„-L2*.
The first equality in (4) is contained in Davis [2], Theorem 3. Moser has
shown (cf. [5]) that
*-*r(?)As variations in (1), we have (with analogous limitations on the range
of?):
(?)
(8)
S
^~7rl
q/2 )'
(P{P~1})
r{p, q) = R'(p, q)~±.
,
Combining these results, we obtain the following statements. The asymptotic relative frequency that a relation on p objects be symmetric is
(10)
* >
R(p)
'
2P<*-V'2
•
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RELATIONAL ASYMPTOTIC RELATIVE FREQUENCIES
259
symmetric irreflexive is
R(p)
2v(p+1>/2
'
reflexive (or irreflexive) is
(12)
~m~~»'
and transitive irreflexive antisymmetric is
(13)
m
._, (^-i)i (2p\
R{p) ~ 2»<*~» \pj'
Some analogous ratios for the asymptotic relative frequencies of structures 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. I t is shown in Ford and Uhlenbeck [3] "that asymptotically, "almost all" graphs are connected, are blocks, and are identity
graphs. The translation of these results to the language of relations 3 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 n[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, Transactions of the American Mathematical
Society, vol. 78 (1955) 445—463.
[5] R. L. W I N E 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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