By Mao L.

This monograph is inspired with surveying arithmetic and physics by means of CC conjecture, i.e., a mathematical technology may be reconstructed from or made via combinatorialization. subject matters coated during this publication comprise basic of mathematical combinatorics, differential Smarandache n-manifolds, combinatorial or differentiable manifolds and submanifolds, Lie multi-groups, combinatorial critical fiber bundles, gravitational box, quantum fields with their combinatorial generalization, additionally with discussions on primary questions in epistemology. All of those are worthy for researchers in combinatorics, topology, differential geometry, gravitational or quantum fields.

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Proof First, choose any element x1 ∈ X. From X \ {x1 }, then choose a second element x2 and from X \ {x1 , x2 } a third element x3 , and so on. Since X is infinite, for any integer n, X \ {x1 , x2 , · · · , xn } can never be empty. Whence, we can always choose an new element xn+1 in the set X \ {x1 , x2 , · · · , xn }. , a countable subset X of X. 1(Dedekind-Peirce) A set X is infinite if and only if it has a bijection with a proper subset of itself. Proof If X is a finite set of cardinal number n, then there is a bijection f : X → {1, 2, · · · , n}.

Therefore, there are (2p − 1)p−2 (p + 1)! such different pairs (5 − 1) and (5 − 2). Notice that each of them defines a district vertex-edge labeled tree of p vertices. Since each vertex-edge labeled tree uniquely defines a pair of there sequences and vice versa. We find the number of vertex-edge labeled trees of order p asserted in this theorem. Similarly, for vertex labeled tress we can also get the number of such trees of order p, which was firstly gotten by Cayley in 1889 shown in the next result.

Whence, we can always choose an new element xn+1 in the set X \ {x1 , x2 , · · · , xn }. , a countable subset X of X. 1(Dedekind-Peirce) A set X is infinite if and only if it has a bijection with a proper subset of itself. Proof If X is a finite set of cardinal number n, then there is a bijection f : X → {1, 2, · · · , n}. If there is a bijection h from X to its a proper subset Y with cardinal number k, then by definition we deduce that k = |Y | = |X| = n. By assumption, Y is a proper subset of a finite set X.

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Combinatorial geometry with application to field theory by Mao L.
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