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225-Notes01

225-Notes01 - Stat 225 Lecture Notes"Set Theory and...

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Stat 225 Lecture Notes “Set Theory and Fundamentals of Probability” Ryan Martin Spring 2008 1 Set Theory Set theory is concerned with mathematical properties of very abstract collections of ob- jects. In this course, we will need only the very basics, which we discuss below. This material is taken primarily from Weiss, Section 1.2. 1.1 Set Notation A set is just a collection of objects, called elements . Some common notations follow. Universal and Empty Sets: It is assumed that there is a universal set, denoted by U or Ω, that contains everything. There is also an empty set, denoted by , that contains nothing. Containment: If x is an element of a set A , we write x A . If x is not an element of A , we write x / A . Subsets: If for two sets A and B , all elements in A are also in B , then we A a subset of B and write A B . If A is not a subset of B , we write A n⊂ B . To verify that A B , one must show that if x A then x B . Sets are often deFned by some property that its elements satisfy. If P ( x ) is some statement regarding x , then { x Ω : P ( x ) } is the set of all x Ω such that P ( x ) is true. ±or example, P ( x ) could be [ x 2]. In this case, { x Ω : P ( x ) } = ( , 2]. Defnition 1.1. Let Ω be a set with subsets A and B . (i) The complement of A is A C = { x Ω : x / A } (ii) The union of A and B is A B = { x Ω : x A or x B } . (iii) The intersection of A and B is A B = { x Ω : x A and x B } . Exercise 1.2. Let Ω = { 1 , 2 , 3 , 4 , 5 , 6 } , A = { 1 , 2 , 3 , 4 } and B = { 2 , 4 , 6 } . Write out the elements in each of A B , A B and B C . 1

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Exercise 1.3. If A = { x : x r } and B = { x : x > s } for s r , describe A C and A B . Those of you familiar with formal logic might see some similarities between the nota- tion there and the set notation introduced above. In particular, ∪ ⇐⇒ “or” , ∩ ⇐⇒ “and” , ( · ) C ⇐⇒ “not” To illustrate this relationship, let P ( x ) and Q ( x ) be statements concerning an element x and let A = { x : P ( x ) } and B = { x : Q ( x ) } . Then A B = { x Ω : P ( x ) or Q ( x ) } A B = { x Ω : P ( x ) and Q ( x ) } A C = { x Ω : not P ( x ) } Defnition 1.4. To sets A and B are said to be disjoint if A B = . In other words, the sets A and B have no elements in common. Exercise 1.5. For an arbitrary set A , give an example of a set B such that A B = . 1.2 Venn Diagrams Venn diagrams are an easy way to visualize sets and set operations. Let Ω be the universal set with subsets A , B , C , etc. Figure 1 shows an example of a Venn diagram for three sets A , B and C .
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225-Notes01 - Stat 225 Lecture Notes"Set Theory and...

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