Mathematical Structure of Singularities at Transitions by H.N. Shirer, R. Wells By H.N. Shirer, R. Wells

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Fully parenthesized proposition: any proposition that can be obtained using the following recursive deﬁnition: each variable is fully parenthesized, if P and Q are fully parenthesized, so are (¬P ), (P ∧ Q), (P ∨ Q), (P → Q), and (P ↔ Q). function f : A → B: a rule that assigns to every object a in the domain set A exactly one object f (a) in the codomain set B. functionally complete set: a set of logical connectives from which all other connectives can be derived by composition. c 2000 by CRC Press LLC fuzzy logic: a system of logic in which each statement has a truth value in the interval [0, 1].

Aim ) ∈ Ai1 × Ai2 × · · · × Aim , there is at most one n-tuple in R that matches (ai1 , ai2 , . . , aim ) in coordinates i1 , i2 , . . , im . composition (of relations): for R a relation from A to B and S a relation from B to C, the relation S ◦ R from A to C such that a(S ◦ R)c if and only if there exists b ∈ B such that aRb and bSc. composition (of functions): the function f ◦ g whose value at x is f (g(x)). compound proposition: a proposition built up from atomic propositions and logical connectives.

Ramanujan left behind several notebooks containing statements of thousands of results, enough work to keep many mathematicians occupied for years in understanding and proving them. Frank Ramsey (1903–1930), son of the president of Magdalene College, Cambridge, was educated at Winchester and Trinity Colleges. He was then elected a fellow of King’s College, where he spent the remainder of his life. Ramsey made important contributions to mathematical logic. What is now called Ramsey theory began with his clever combinatorial arguments to prove a generalization of the pigeonhole principle, published in the paper “On a Problem of Formal Logic”.