INDEX OF SYMBOLS Set-theoretic notions x G X x is a member of X, 1 x $L X x is not a member of X, 1 0 empty set, 3 {#, y,... } set of elements x, y,..., 7 {x : /[#]} set of all elements satisfying ip, 1 (#, y) ordered pair of x and y, 1 (a,i : i n) sequence of the elements ao,..., an-i 12 a*6 concatenation of the sequences a and 6, 1 I C 7 X is a subset of Y, 8 56 (X) set of all subsets of X, 1 I U 7 union of X and y , 27 i n F intersection of X and y , 8 (JX union of all members of X, 4 P|X intersection of all members of X, 63 Uie/^i union of family of sets X{, 63 Cliei^i intersection of family of sets X\, 63 X ~ y difference of X and y , 1 ~ y complement of Y in a fixed universe, 1 |X| cardinality of the set X, 1 X x y set of ordered pairs (x, y) with x G X and y G F , 1 x y set of all functions from X to Y, 1 IlnGu Xn/F ultraproduct of X n modulo the ultrafilter F, 42 /[X] image of X under the function / , 1 / _ 1 [ y ] inverse image of Y under the function / , 1 P,Q,R,S,T usually denote binary relations, 1 Iv identity relation on the set U, 1 Dv diversity relation on the set [/, 1 / Iu when U is understood, 1 D Dv when U is understood, 1 R\S relational composition of R and 5, 1 F o G functional composition of F and G, 1 R~l relational inverse of R, 1 115
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