Nonlinear Control Sy.. - Free

Chapter 2
Mathematical Preliminaries
This chapter collects some background material needed throughout the book. As the
material is standard and is available in many textbooks, few proofs are offered. The
emphasis has been placed in explaining the concepts, and pointing out their importance
in later applications. More detailed expositions can be found in the references listed at the
end of the chapter. This is, however, not essential for the understanding of the rest of the
book.
2.1 Sets
We assume that the reader has some acquaintance with the notion of set. A set is a collection
of objects, sometimes called elements or points. If A is a set and x is an element of A, we
write x E A. If A and B are sets and if every element of A is also an element of B, we say
that B includes A, and that A is a subset of B, and we write A C B or B D A. As it has
no elements, the empty set, denoted by 0, is thus contained in every set, and we can write
0 C A. The union and intersection of A and B are defined by
AUB = {x:xEAorxEB} (2.1)
AnB = {x:xEAandxEB}. (2.2)
Assume now that A and B are non empty sets. Then the Cartesian product A x B
of A and B is the set of all ordered pairs of the form (a, b) with a E A and b E B:
AxB={(a,b):aeA, andbEB}. (2.3)
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