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Mathematics LibreTexts

7.1: A- Basic Notation and Terminology

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We quickly review some basic notation used in this book that is perhaps not described elsewhere. We use C, R for complex and real numbers, and i for imaginary unit (a square root of 1). We use N={1,2,3,} for the natural numbers, N0={0,1,2,3,} for the zero-based natural numbers, and Z for all integers. When we write Cn or Rn we implicitly mean that n1, unless otherwise stated.

We denote set subtraction by AB, meaning all elements of A that are not in B. We denote complement of a set by Xc. The ambient set should be clear. So, for example, if XC naturally, then Xc=CX. Topological closure of a set S is denoted by ¯S, its boundary is denoted by S. If S is a relatively compact subset of X (its closure in X is compact) or compact, we write S⊂⊂X.

A function with domain X and codomain Y we denote by f:XY. The direct image of S by if is f(S). The notation f1 for the inverse image of sets and single points. When f is bijective (one-to-one and onto), we use f1 for the inverse mapping. So f1(T) for a set TY denotes the points of X that f maps to T. For a point q, f1(q) denotes the points that map to q, but if the mapping is bijective, then it means the unique point mapping to q. To define a function without giving it a name, we use xF(x),

where F(x) would generally be some formula giving the output. The notation f|S is the restriction of f to S: a function f|S:SY such that f|S(x)=f(x) for all xS. A function f:UC is compactly supported if the support, that is the set ¯{pU:f(p)0}, is compact. If f(x)=g(x) for all x in the domain, we write fg,
and we say that f and g are identically equal. The notation fg
denotes the composition defined by xf(g(x)).

To define X to be Y rather than just show equality, we write Xdef=Y.


This page titled 7.1: A- Basic Notation and Terminology is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Jiří Lebl via source content that was edited to the style and standards of the LibreTexts platform.

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