LEC_EAS305_F11_0831

# LEC_EAS305_F11_0831 - Set Theory Experiments, Sample...

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Set Theory Experiments, Sample Spaces, and Events Probability Basics Five Theorems Fall 2011 EAS 305 Lecture Notes Prof. Jun Zhuang University at Buﬀalo, State University of New York August 31 . .. 2011 Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 1 of 26

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Set Theory Experiments, Sample Spaces, and Events Probability Basics Five Theorems Agenda for Today 1 Set Theory Deﬁnition and Notations Laws of Operation 2 Experiments, Sample Spaces, and Events Deﬁnition and One Example Events Remarks and Examples 3 Probability Basics Deﬁnition and Notations Frequentist view One Example 4 Five Theorems Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 2 of 26
Set Theory Experiments, Sample Spaces, and Events Probability Basics Five Theorems Deﬁnition and Notations Laws of Operation Deﬁnition and Notations Deﬁnition: A set is a collection of objects. Members of a set are called elements. Notation: A , B , C , ... for sets a , b , c , ... for elements for membership, e.g., x A / for non-membership, e.g., x / A U is the universal set (i.e., everything) is the empty set Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 3 of 26

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Set Theory Experiments, Sample Spaces, and Events Probability Basics Five Theorems Deﬁnition and Notations Laws of Operation Examples of Sets A = { 1 , 2 , ..., 10 } . 2 A , 49 / A B = { basketball , baseball } C = { x | 0 x 1 } (“ | ” means “such that”) D = { x | x 2 = 9 } = 3 } = { 3 , - 3 } E = { x | x ∈ < , x 2 = - 1 } = ( < is the real line) Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 4 of 26
Set Theory Experiments, Sample Spaces, and Events Probability Basics Five Theorems Deﬁnition and Notations Laws of Operation Examples of Sets A = { 1 , 2 , ..., 10 } . 2 A , 49 / A B = { basketball , baseball } C = { x | 0 x 1 } (“ | ” means “such that”) D = { x | x 2 = 9 } = 3 } = { 3 , - 3 } E = { x | x ∈ < , x 2 = - 1 } = ( < is the real line) Toys. .. Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 5 of 26

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Set Theory Experiments, Sample Spaces, and Events Probability Basics Five Theorems Deﬁnition and Notations Laws of Operation Deﬁnition and Properties Deﬁnition: If every element of set A is an element of set B then A is a subset of B , i.e., A B Deﬁnition: A = B iﬀ (if and only if) A B and B A Properties: ∅ ⊆ A A U A A A B and B C (implies) A C Remark: Order of element listing is immaterial, e.g, { a , b , c } = { b , c , a } Prof. Jun Zhuang Fall 2011 EAS 305 Lecture Notes Page 6 of 26
Experiments, Sample Spaces, and Events Probability Basics Five Theorems Deﬁnition and Notations Laws of Operation More deﬁnitions Complement of A with respect to U is ¯ A ≡ { x | x U and x / A } Intersection of A and B is A B ≡ { x | x A and x B } . Union of

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## LEC_EAS305_F11_0831 - Set Theory Experiments, Sample...

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