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3. It is clear that M \ N # N \ M , unless M = N , in which case M \ N = 8. Fig. 3 Difference of two sets 37 Sets Given three sets L , M and N we can form two new sets A = ( L u M ) u N and B = L u ( M U N ) . For example, for B we form first the union M U N = Q and then use Q to form L U Q. It is easy to see that each of the new sets A,B consists of those elements belonging to at least one of the sets L , M , N . Hence ( L U M ) U N = L U ( M U N ) and we write simply L U M U N . The same convention is used for intersections; for example, the intersection of four sets A,B,C , D may be denoted simply by A n B n C n D.

3 Families of sets Instead of saying that A is a set of sets we prefer to say that A is a system or a family of sets. Later we shall encounter situations where to each element l of a set L there is associated a set CZ. In such a situation we say that the family of sets Cl is indexed by the set L. It is natural to denote this indexed family by {Cl;l E L } . If {Cl;l E L } is an indexed family then by Cl we understand the union of the family Cl, that is, the set of all u 1EL elements belonging to at least one set Cl.

1 Expression sequences, lasts An expression is a basic entity in Maple; roughly speaking, it is a string of characters which can be entered into Maple. An expression sequence is an ordered n-tuple of expressions separated by commas. Hy%, set, x+l/x, 1, 1 is an expression sequence. An expression sequence enclosed in square brackets constitutes a list. Order and repetition of expressions in lists or expression sequences do matter: [I, 21 , [2, 13 and [I, 2, 11 are all distinct. Sets, lists and expression sequences are all similar: the difference lies in the way Maple handles them.

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Acourse of pure mathematics by Hardy

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