1Mathematical_Review

1Mathematical_Review - Chapter 1 Mathematical Review Set...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 2 CHAPTER 1. MATHEMATICAL REVIEW same elements. If x and y are different objects then we write x = y . Also, we can express the fact that S and T are different 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 different from T . In fact, S = T if and only if S ⊂ T and T ⊂ S . If S ⊂ T and S is different 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 } ....
View Full Document

This note was uploaded on 03/30/2010 for the course ECEN 303 taught by Professor Chamberlain during the Fall '07 term at Texas A&M.

Page1 / 7

1Mathematical_Review - Chapter 1 Mathematical Review Set...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online