By P. K. Jain, Ahmed Khalid

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IDom/, := f 0 (', b) IDom / . )(y) for all (x, y) E Domf. j): Ai ~ QAi by the rule .. )')()) Z I' {c if 1\ {j} } . E . j) := (a I[\{il. j ). j) IDom / . 25) for all x E QAi. j)(xi) = f(x) Let S, T and C be sets. ))(y) := f(x, y) for all xES, yET. 28) Thus, we obtain the identification Map(S x T,C) ~ Map (S,Map(T,C)). 29) 20 Chapter 0 Basic Mathematics Notes 04 (1) The notation U(Ai liE I) for the union of the family (Ai liE I) is used when it occurs in the text rather than in a displayed formula.

This means that one uses the term "addition" for the combination, one writes a + b := cmb(a,b) and calls it the "sum" of a and b, one calls the neutral "zero" and denotes it by 0, and one writes -a := rev(a) and calls it the "opposite" of a. The abbreviation a - b := a + (-b) is customary. The number sets Nand P are additive monoids while l, Q, R, and C are additive groups. 13) S + T:= cmb>(S x T) = {s + tis E s, t E T}. and call it the member-wise sum of Sand T. If t EM, we abbreviate S + t:= S + {t} = {t} + S =: t + S.

The set of all invertible mappings from a set 8 to itself is denoted by Perm 8 and its members are called permutations of 8. Let 8 and T be sets. For each x E 8, we define the evaluation at x, ev:z: : Map (8, T) -+ T, by the rule = eV:z:(I) := I(x) for all I E Map (8, T). 10) Assume now that a subset F of Map (8, T) is given. Then the restriction ev:z: IF is also called the evaluation at x and is simply denoted by ev:z: if the context makes clear what F is. We define the mapping evF : 8 -+ Map (F, T) by (evF(x))(I) := eV:z:(I) = I(x) for all x E 8, IE F.