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Binary Relations on the Power Set of an N-Element Set

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Binary Relations on the Power Set of an N-Element Set 1 2 Journal of Integer Sequences, Vol. 12 (2009), 3 Article 09.2.6 47 6 23 11 Binary Relations on the Power Set of an n-Element Set Ross La Haye 955 Coppens Road Green Bay, WI 54303 USA [email protected] Abstract We define six binary relations on the power set of an n-element set and describe their basic structure and interrelationships. An auxiliary relation is noted that will assist in determining the cardinalities of each. We also indicate an eighth relation that may be of interest. We conclude the second section by computing several quantities related to walks in the graph of the sixth relation. In the third section we turn our attention to the basic structure and cardinalities of the auxiliary relation noted in section two and several additional relations. We also compute seven sums associated with these relations and indicate connections four relations have with Wieder’s conjoint and disjoint k-combinations. 1 Introduction Combinatorial interest has generally focused on the enumeration of various types of relations or associated structures such as topologies and graphs. In this paper we will be interested in the generally more tractable problem of defining and enumerating several binary relations on the power set of an n-element set and computing various quantities associated with them. We will conclude by indicating connections four relations have with Wieder’s conjoint and disjoint k-combinations. The motivation here is to suggest some ways in which sets of k- subsets of some set may be studied under the rubric of asymmetric k-ary relations on the same. To the author’s knowledge, the results of this paper or related results do not appear in the existing literature, except when so noted. 1 2 The six initial relations 2.1 Definitions and structure Let P (A) be the power set of an n-element set A1. Then B = P (A) × P (A) is just the Cartesian product of P (A) with itself and the following binary relations on P (A) are, of course, just subsets of B: R0 = {(x,y): x,y ∈ P (A),x ⊆ y ∨ y ⊆ x}, R1 = {(x,y): x,y ∈ P (A),x ⊂ y ∨ y ⊂ x}, R2 = {(x,y): x,y ∈ P (A),x ≺ y ∨ y ≺ x}, R3 = {(x,y): x,y ∈ P (A),x ⊆ y}, R4 = {(x,y): x,y ∈ P (A),x ⊂ y}, and R5 = {(x,y): x,y ∈ P (A),x ≺ y}. Note that x ≺ y denotes that x ⊂ y and |y|−|x| =1[2]. As R3 has been defined to be a partial order relation on P (A), this just means that R5 is the corresponding covering relation [2]. R4 is the corresponding strict order [2]. Also, observe that B is a reflexive, symmetric, and transitive relation on P (A) and so is an equivalence relation on the same. R0 R1 R2 R3 R4 R5 reflexive yes no no yes no no irreflexive no yes yes no yes yes symmetric yes yes yes no no no asymmetric no no no no yes yes antisymmetric no no no yes yes yes transitive no no no yes yes no intransitive no no yes no no yes Table 1: Basic properties of R0 – R5 Tables 1, 2, and 3 list some basic information with regard to R0 – R5. While entries =, ⊆, and ⊇ in Table 3 are easily enough understood, we explain an entry of ∩ as follows. Given a row relation Rr and a column relation Rc in Table 3, an entry of ∩ indicates that, while Rr 6⊆ Rc and Rc 6⊆ Rr, we still have that Rr ∩ Rc =6 ∅. Examples. With A = {1, 2}, P (A)= {∅, {1} , {2} , {1, 2}}, and R4 = {(∅, {1}), (∅, {2}), (∅, {1, 2}), ({1} , {1, 2}), ({2} , {1, 2})}, R5 = {(∅, {1}), (∅, {2}), ({1} , {1, 2}), ({2} , {1, 2})}. 1For convenience we may assume that A = {1, 2, 3,...,n}. 2 R0 R1 R2 R3 R4 R5 reflexive closure R0 R3 reflexive reduction R1 R4 symmetric closure R0 R1 R2 symmetric reduction R3 R4 R5 transitive closure R4 transitive reduction R5 Table 2: Closures and reductions of R0 – R5 R0 R1 R2 R3 R4 R5 R0 = ⊇⊇⊇⊇⊇ R1 ⊆ = ⊇ ∩ ⊇ ⊇ R2 ⊆ ⊆ = ∩ ∩ ⊇ R3 ⊆ ∩ ∩ = ⊇ ⊇ R4 ⊆ ⊆ ∩ ⊆ = ⊇ R5 ⊆⊆⊆⊆⊆ = Table 3: Inclusion and intersection relationships, R0 – R5 Define the partial order relation E = {(p,q): p,q ∈ {R0,R1,...,R5} ,p ⊆ q} so that G may be defined to be the (directed) graph of the covering relation corresponding to E. Then most of the information in Tables 1, 2, and 3 is illustrated in Figure 1. Observe that we have labelled the edges of G in the diagram to indicate the closure and reduction relationships that obtain between the vertices of this graph, i.e., R0,R1 etc. For example, an edge labelled s and directed from vertex u to vertex v indicates that u is the symmetric reduction of v (and v the symmetric closure of u); we can also deduce from this that v is a symmetric relation. Similar observations hold for the other two edge labels. Finally, note that if a walk exists from u to v, then u ⊆ v. However, if there is neither a walk from u to v nor a walk from v to u, then, while u 6⊆ v and v 6⊆ u, it is still the case that u ∩ v =6 ∅. R 0 B ||> ` BB s || BB || r BB || B R R 3 B 1 B ` BB ||> ` BB BB s || BB r BB || BB B || B R R 4 B 2 ` BB ||> BB s || t BB || B || R5 Figure 1: Hasse diagram of G 3 2.2 The cardinalities of R0 – R5 In determining the cardinalities of R0 – R5 it will be useful to recall [1] that |P (A)| =2n. (1) To see this, consider that in enumerating a subset of A, i.e., an element of P (A), we decide on a case by case basis which elements of A to include in the subset. As there are 2 outcomes possible for each of the elements of A – include in the subset or exclude from the subset – and n elements of A, there are 2n subsets of A altogether. Now recall that the (x,y) ∈ R3 are such that x ⊆ y. It is not hard to see that enumerating each subset of each subset of A will enumerate the elements of R3. We may do so by deciding on a case by case basis which elements of A to 1. include in both x and y, 2. exclude from both x and y, or 3. exclude from x and include in y. As there are 3 outcomes possible for each of the n elements of A, we see that n |R3| =3 . (2) −1 The cardinalities of R0, R1, and R4 are determined as follows. Let Ri denote the inverse relation of Ri [3]. Then, define R11 = {(x,y): x,y ∈ P (A),x = y} = {(x,x): x ∈ P (A)}. It should be clear that R11 ⊂ R3. Moreover, it is obvious that n |R11| =2 . (3) It then follows that n n |R0| =2 · 3 − 2 , (4) −1 as, by definition, R0 = R3 ∪ R3 , and R11 should only be counted once in this union. Similarly, we see that n n |R4| =3 − 2 , (5) as, by definition, R4 = R3 \ R11. Furthermore, n n |R1| =2 · (3 − 2 ), (6) −1 as, by definition, R1 = R4 ∪ R4 . Recall that for each (x,y) ∈ R5, we have that x ⊂ y and |y|−|x| = 1. This suggests the following enumeration of this relation. First, select one of the n elements of A to add to x (to get y). Then x is constructed from the remaining elements of A, and this can be done in 2n−1 ways. As y is now completely determined, we have that n−1 |R5| = n · 2 . (7) 4 It is then easy to see that n |R2| = n · 2 , (8) −1 as, by definition, R2 = R5 ∪ R5 . Before concluding here we note one additional relation that may be of interest: R6 = R4 \ R5. R6 is an irreflexive, asymmetric, antisymmetric, and transitive relation and so a strict order. We also note that this relation is a proper subset of R0, R1, R3, and R4. Finally, that n n n−1 |R6| =3 − 2 − n · 2 (9) is easily deduced from Eqns. (5) and (7). Relation Cardinality Sequence Sequence Sequence n n R0 2 · 3 − 2 A027649(n) A090888(n, 3) n n R1 2 · (3 − 2 ) A056182(n) n R2 n · 2 A036289(n) n R3 3 A000244(n) A090888(n, 2) A112626(n, 0) n n R4 3 − 2 A001047(n) A090888(n, 1) A112626(n, 1) n−1 R5 n · 2 A001787(n) A090802(n, 1) n n n−1 R6 3 − 2 − n · 2 A066810(n) A112626(n, 2) Table 4: The cardinalities of R0 – R6 Table 4 lists the cardinalities of R0 – R6 with references to the corresponding integer sequences in Sloane’s Encyclopedia [5]. 2.3 Walks and lengths of walks in the graph of R5 Let H be the (directed) graph of R5 and recall that P (A) is the power set of an n-element set A. Also, let C(n, k) denote the binomial coefficient, i.e., the number of ways to select k elements from an n-element set. We begin by counting the number of k-length walks w(n, k) in H [3]. As the vertices of H are just elements of P (A), a little thought shows that a k-length walk in H begins at some x ∈ P (A) and adds k elements of A not in x to x, where the order of addition is important.
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