1Mathematical_Review

1Mathematical_Review - Chapter 1 Mathematical Review Set...

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Chapter 1 Mathematical Review Set theory is now generally accepted as the foundation of modern mathemat- ics, and it plays an instrumental role in the treatment of probability. Un- fortunately, a simple description of set theory can lead to paradoxes, while a rigorous axiomatic approach is quite tedious. As such, we assume that the meaning of a set as a collection of objects is intuitively clear and proceed to define relevant notation and operations. This standpoint is known as naive set theory. 1 2 3 4 5 6 7 Figure 1.1: This is an illustration of a generic set and its elements. A set is a collection of objects, which are called the elements of the set. If an element x belongs to a set S , we express this fact by writing x S . If x does not belong to S , we write x / S . We use the equality symbol to denote logical identity . For instance, x = y means that x and y are symbols denoting the same object. Similarly, the equation S = T states that S and T are two symbols for the same set. In particular, the sets S and T contain precisely the 1
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2 CHAPTER 1. MATHEMATICAL REVIEW same elements. If x and y are di ff erent objects then we write x = y . Also, we can express the fact that S and T are di ff erent sets by writing S = T . A set S is a subset of T if every element of S is also contained in T . We express this relation by writing S T . Note that this definition does not require S to be di ff erent from T . In fact, S = T if and only if S T and T S . If S T and S is di ff erent from T , then S is a proper subset of T , which we indicate by S T . There are many ways to specify a set. If the set contains only a few elements, one can simply list the objects in the set; S = { x 1 , x 2 , x 3 } . The content of a set can also be enumerated whenever S has a countable number of elements, S = { x 1 , x 2 , . . . } .
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