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Set Theory and Lattices Appendix A SET THEORY AND LATTICES The objective of the Appendices is to present the main definitions and propositions of some of the mathematical structures used in the text. With a few exceptions, no attempt was made to present proofs of the propositions, for otherwise the length of the Appendices would be greater than the main text. Any serious student who needs to improve his knowledge about the topics treated below must consult mathematical texts as, e.g., [1-6] and also the following excellent texts on quantum theory[7-10]. A.1 MAIN DEFINITIONS There are two primitive concepts in set theory, the notion of set itself1 and the notion of pertinence as used, e.g., in sentences like "x is an element of the set S." We write xES if x is a member of a given set S and write x ~ S otherwise. In general a set S is determined by some property P shared by its members and we denote this fact writing S = {xIP(x)}. definition 1. Given any property P, the empty (or null, or void) set is the set 0= {xIP(x) is false}. (A. 1) definition 2. Let A and B be two sets such that Vx E A => x E B. We say that A is a subset of B and denote this fact, writing A ~ B or B ;2 A. If it happens that A ~ B and B ;2 A, then both sets are equal and we write A = B. If A ~ B or A :f:. B we say that A is a proper subset of B and write sometimes A C B. The symbol ~ (and also C) is called set theoretical inclusion. We observe that given any set S we have 0 ~ S (or better 0 C S). definition 3. Let A ~ S. The mapping A -+ S, (A.2) 1 if x E A, XA(X) = 0 if x ~ A. 187 188 NONLOCALITY IN QUANTUM PHYSICS is called the characteristic function of the set A. definition 4. Given two sets A and B, their union is the set denoted A U B such that Au B = {xix E A or x E B}. (A.3) definition 5. Given two sets A and B, their intersection is the set denoted A n B such that AnB = {xix E Aandx E B}. (A.4) If An B = 0, then A and B are said to be disjoint. definition 6. The complement of A ~ S is the set denoted AC (or S \ A) such that A C = {x ~ A, XES}. (A.5) definition 7. Let A, B ~ S. The difference of A and B is the set denoted A - B such that A - B = An BC= {xix E A and x ~ B}. (A.6) The difference of A and B is also called the relative complement of B in A. definition 8. The symmetric difference of A and B (A, B ~ S) is the set denoted Al:::,.B such that Al:::,.B = (A - B) U (B - A). (A.7) definition 9. Let J be a set whose elements we will call indexes. Given a set X, an indexedfamity of elements of X with indexes in J is a mapping x : J -t X, a t-+ XO. The indexed family is denoted {Xo}oEJ or simply {xo } when it is clear who the set J is. When J = N = {1, 2, ... }, the indexed family is said to be a sequence of elements ofX. definition 10. An indexed family {Ao} of subsets of A, with indexes in J is said to be a partition of A if \la, (3 E J, a :I (3, Ao n A,B = 0 and UOEJAo = A definition 11. The union and the intersection of the Ao are respectively the sets UOEJAo {xix E Ao for at least one E J}, noEJAo = {xix E Ao for all a E J}. (A.8) definition 12. Given a set S, the class of all its subsets is said to be the power set of S and is denoted by 28 . definition 13. The Cartesian product of the n sets AI, A2 , . .. An is the set When Ai = A, i = 1,2, ... n we write the Cartesian product as An. APPENDIX A: SET THEORY AND LATTICES 189 A.2 PO SETS definition 14. A (binary) relation in a set 8 is a specified subset R ~ 8 x 8. If (x, y) E R we say that x stands in relation with y and write xRy. definition 15. Let R ~ 8 x 8 be a relation in 8. It is said to be: (i) reflexive if xRx, 'Vx E S, (ii) symmetric if xRy ::::} yRx, (iii) transitive if xRy, yRz ::::} xRz, (iv) antisymmetric if xRy and yRx ::::} x = y. definition 16. An equivalence relation in a set 8 is a relation in 8 which is reflexive, symmetric and transitive. We often use the symbol x'" y to denote that x, y E 8 are equivalent. proposition 17. An equivalence relation in S leads to a unique partition of S. Conversely, any given partition of 8 defines an equivalence relation in 8. definition 18. Let'" be an equivalence relation in S. Let xES be an arbitrary but fixed element. Consider the collection of all y such that x '" y. This collection denoted [xl ~ 8 is called the equivalence class of x. We have, [xl = {y E 8 and y '" x}. (A 10) definition 19. Let E ~ S x 8 be an equivalence relation in 8. The quotient set of S modulo E, denoted by SI E is the class of all distinct equivalence classes induced in 8 by E. We have, 81E = {[xli x E 8}. (All) definition 20. The mapping 71' : S -t S IE; x t-+ [x Jis called the canonical mapping (or projection) of S onto SI E. definition 21. A relation in a set S is called an order relation if it is reflexive, anti symmetric and transitive. The symbol::; is used to denote in what follows the order relation. If x, y, z E 8 we have, (i) x < x, 'Vx E 8, (ii) x < y and y ::; x ::::} x = y, (iii) x < y and y ::; z ::::} x ::; z. (AI2) Sometimes instead of x ::; y (which reads x is less than or equal to y, or x is contained in y) we write y ~ x. If x::; y and x f:. y we write x < y. When x < yand there is no z such that x < z < y we say that y covers x and write x -< y. definition 22. Given a set 8 and an order relation::; in 8, the pair (8,::;) is called a poset (a short form for partially ordered set).2 definition 23. Let (8,::;) be a po set. If there exists 0 E 8 such that 0 ::; x, 'Vx E 8, then 0 is said to be the first (or smallest, or least) element of 8. If there exists 1 E 8 such that x ::; 1, 'Vx E 8, then 1 is said to be the last (or largest, or greatest) element of S. 190 NONLOCALITY IN QUANTUM PHYSICS proposition 24. The first element of 5 when it exists is unique. The last element of 5 if it exists, is unique. definition 25. Let (5, ::;) be a poset. m E 5 is called a minimal element of 5 if there is no element in 5 which is strictly smaller than m, i.e., if x ::; m, then x = m. An element m E 5 is said to be the maximal element of 5 if there is no element of 5 that is strictly greater than m, i.e., if m ::; x, then m = x. definition 26. Let (5, ::;) be a poset and A ~ 5 (eventually A = 5). l E 5 is said to be a lower bound of of A if and only if Va E A :::} l ::; a. An element u E 5 is called an upper bound of A if and only if Va E A :::} u :::: a. It is important to have in mind that A ~ 5 need not have necessarily any lower bound and even if it does, let us say b, in general b ¢ A. In this way it is clear that A may have many different lower bounds. An analogous comment is valid regarding upper bounds. definition 27. Let (5,::;) be a poset and A ~ 5. Denote by AL the set of all lower bounds of A. If AL has a last (greatest) element, then it is called the infimum (or the greatest lower bound) and is denoted inf(A). Denote the set of all upper bounds of A by Au. If Au has a first (smaller) element then it is called the supremum (or least upper bound) denoted sup(A). The notations VaEAa and AaEAa are often used respectively for inf(A) and sup(A). When A is a finite countable (infinite countable) set we write for the infimum Vi=l a (V~l a). For the supremum we write A;;' 1 a (A~l a). proposition 29. Let (5, ::;) be a poset and x, y, z E 5. Then, (i) inf{x,{y,z}} inf{x,y,z} (ii) sup{ x, {y, z}} sup{x,y,z} A.3 LATTICES definition 30. A poset C is called a lattice if: (i) :1 0 E C, :1 1 E C and 0 =j:. 1, (ii) each pair x, y E C, x =j:. y, has a supremum (also called union, join or dis­ junction) and denoted by x V y and an infimum (also called intersection, meet or conjunction) denoted by x A y.
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