A basis of identities of the Lie algebra s(2) over a finite by Semenov K.N.

By Semenov K.N.

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Two elements A and B are connected by a nonhorizontal (usually straight) line if A ≤ B and there does not exist any element C (≠ A, B) such that A ≤ C ≤ B (in the diagram, A is situated below B). For example, the Hasse diagram of the lattice A ≤ B ≤ C is shown in Fig. 1. C B A Fig. 1. Hasse diagram representation of the lattice A ≤ B ≤ C. 4 Hasse Diagram 49 The Hasse diagrams of the lattices that can be formed using n = 1, 2, 3, 4, 5 elements are shown below. Any 1-element set is a lattice; its Hasse diagram is a single point: •.

2. 3 • “I am reading this text ȁ It is raining” is a proposition, and its truth value can be assigned by the reader. • “I am thinking to myself ȁ A bicycle has two wheels” is a proposition (the reader can assign a truth value to it), albeit that one would rarely link its two constituent propositions into one sentence in everyday speech. 4 Disjunction Given two propositions P, Q, the proposition denoted by P V Q (expressed as “P or Q”) is called a disjunction. 3). Thus, P V (¬P) is always true (law of the excluded third).

9) k =0 where C nk denotes the combinations of n taken by k. 15: |{∅, {thought}, {ape}, {quantum}, {thought, ape}, {thought, quantum}, {ape, quantum}, {thought, ape, quantum}}| = 23 = 8. 6 for an analogy in mathematical logic): CA(B ∩ C) = CA(B) ∪ CA(C). CA(B ∪ C) = CA(B) ∩ CA(C). 3 Elements of Relations Theory The aim of this section is to present the major concepts of relations theory, with an emphasis on ordering and equivalence relations (as applied in IR). However trivial the word “order” might sound, it is all important in theoretical sciences as well in practical applications.

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