By Peter L Duren; Richard Askey; Uta C Merzbach; Harold M Edwards
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Extra info for A Century of mathematics in America
Furthermore, let m denote the number of edges of G, and let f denote the number of disjoint regions in R2 whose boundaries are the edges of G. Then, n − m + f = 2. Consequently, if n ≥ 3, then m ≤ 3(n − 2). 1. The following identities hold: i) Let 0 ≤ k ≤ n. Then, ii) Let 1 ≤ k ≤ n. Then, k n k = n k =n n . n−k n−1 . k−1 15 PRELIMINARIES iii) Let 2 ≤ k ≤ n. Then, n k k(k − 1) = n(n − 1) n−2 . k−2 iv) Let 0 ≤ k < n. Then, n k (n − k) =n n−1 . k v) Let 1 ≤ k ≤ n. Then, n k = n−1 n−1 + . k k−1 vi) Let 0 ≤ m ≤ k ≤ n.
R R Furthermore, if x1 ≤ x2 and x2 ≤ x3 , then, since R is transitive, it follows that R x1 ≤ x3 . 9. Let “≤” be a partial ordering on X. Then, the following terminology is deﬁned: i) Let S ⊆ X. Then, y ∈ X is a lower bound for S if, for all x ∈ S, it follows R that y ≤ x. ii) Let S ⊆ X. Then, y ∈ X is an upper bound for S if, for all x ∈ S, it follows R that x ≤ y. iii) Let S ⊆ X. Then, y ∈ X is the least upper bound lub(S) for S if y is an upper bound for S and, for all upper bounds x ∈ X for S, it follows that R y ≤ x.
Then, “⊂” and “⊆” are transitive relations on X, and “⊆” is a partial ordering on X. 1. Let G = (X, R) be a graph. Then, the following statements hold: i) R is the graph of a function on X if and only if every node in X has exactly one child. Furthermore, the following statements are equivalent: ii) R is the graph of a one-to-one function on X. iii) R is the graph of an onto function on X. iv) R is the graph of a one-to-one and onto function on X. v) Every node in X has exactly one child and not more than one parent.