DS_Lecture_6.pdf - Lecture 6 Introduction to Sets Dr Chengjiang Long Computer Vision Researcher at Kitware Inc Adjunct Professor at SUNY at Albany Email

DS_Lecture_6.pdf - Lecture 6 Introduction to Sets Dr...

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Lecture 6: Introduction to Sets Dr. Chengjiang Long Computer Vision Researcher at Kitware Inc. Adjunct Professor at SUNY at Albany. Email: [email protected]
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C. Long Lecture 6 September 13, 2018 2 ICEN/ICSI210 Discrete Structures Outline Definitions: Set, Element Terminology and Notation Proving Equivalences Power Set Tuples Cartesian Product Quantifiers
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C. Long Lecture 6 September 13, 2018 3 ICEN/ICSI210 Discrete Structures Outline Definitions: Set, Element Terminology and Notation Proving Equivalences Power Set Tuples Cartesian Product Quantifiers
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C. Long Lecture 6 September 13, 2018 4 ICEN/ICSI210 Discrete Structures Introduction (1) We have already implicitly dealt with sets Integers ( Z ), rationals ( Q ), naturals ( N ), reals ( R ), etc. We will develop more fully The definitions of sets The properties of sets The operations on sets Definition : A set is an unordered collection of (unique ) objects Sets are fundamental discrete structures and for the basis of more complex discrete structures like graphs
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C. Long Lecture 6 September 13, 2018 5 ICEN/ICSI210 Discrete Structures Introduction (2) Definition : The objects in a set are called elements or members of a set. A set is said to contain its elements Notation, for a set A: x Î A: x is an element of A $\in$ x Ï A: x is not an element of A $\notin$
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C. Long Lecture 6 September 13, 2018 6 ICEN/ICSI210 Discrete Structures Outline Definitions: Set, Element Terminology and Notation Proving Equivalences Power Set Tuples Cartesian Product Quantifiers
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C. Long Lecture 6 September 13, 2018 7 ICEN/ICSI210 Discrete Structures Terminology (1) Definition : Two sets, A and B, are equal is they contain the same elements. We write A=B. Example: {2,3,5,7}={3,2,7,5}, because a set is unordered Also, {2,3,5,7}={2,2,3,5,3,7} because a set contains unique elements However, {2,3,5,7} ¹ {2,3} $\neq$
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C. Long Lecture 6 September 13, 2018 8 ICEN/ICSI210 Discrete Structures Terminology (2) A multi-set is a set where you specify the number of occurrences of each element: {m 1 × a 1 ,m 2 × a 2 ,…,m r × a r } is a set where m 1 occurs a 1 times m 2 occurs a 2 times m r occurs a r times In Databases, we distinguish A set: elements cannot be repeated A bag : elements can be repeated
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C. Long Lecture 6 September 13, 2018 9
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