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By Eiichi Bannai

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Extra resources for Algebraic Combinatorics I: Association Schemes (Mathematics lecture note series) (Bk. 1)

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Big-oh notation: f is O(g), written f = O(g), if there are constants C and k such that |f (x)| ≤ C|g(x)| for all x > k. bijection (or bijective function): a function that is one-to-one and onto. bijective function: See bijection. c 2000 by CRC Press LLC binary relation from a set A to a set B: any subset of A × B. , a subset of A × A. body of a clause A1 , . . , An ← B1 , . . , Bm in a logic program: the literals B1 , . . , Bm after ←. cardinal number (or cardinality) of a set: for a finite set, the number of elements; for an infinite set, the order of infinity.

He studied mathematics on his own and soon began producing results in combinatorial analysis, some already known and others previously unknown. At the urging of friends, he sent some of his results to G. H. Hardy in England, who quickly recognized Ramanujan’s genius and invited him to England to develop his untrained mathematical talent. During the war years from 1914 to 1917, Hardy and Ramanujan collaborated on a number of papers, including several dealing with the theory of partitions. Unfortunately, Ramanujan fell ill during his years in the unfamiliar climate of England and died at age 32 soon after returning to India.

Xn }, then (∃x ∈ D)P (x) is true if and only if P (x1 ) ∨ · · · ∨ P (xn ) is true. 5. Adjacent universal quantifiers [existential quantifiers] can be transposed without changing the meaning of a logical statement: (∀x)(∀y)P (x, y) ⇔ (∀y)(∀x)P (x, y) (∃x)(∃y)P (x, y) ⇔ (∃y)(∃x)P (x, y). 6. Transposing adjacent logical quantifiers of different types can change the meaning of a statement. ) 7. x)P (x) ⇔ ¬(∃x)P (x) ∨ (∃y)(∃z)[(y = z) ∧ P (y) ∧ P (z)]. 8. Every quantified statement is logically equivalent to some statement in prenex normal form.

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