Chapter 02

Chapter 02 - 2.1.1 Basic Terminology 2.1.2 Set Operations...

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2.1.1 Basic Terminology 2.1.2 Set Operations 2.1.3 Sets of Sets Chapter 2: Background 2.1 Review of Set Theory B. Champagne 1 1 Department of Electrical & Computer Engineering McGill University September 4, 2009 ECSE 305 2.1 Set theory
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2.1.1 Basic Terminology 2.1.2 Set Operations 2.1.3 Sets of Sets Outline 2.1.1 Basic Terminology 2.1.2 Set operations 2.1.3 Sets of Sets ECSE 305 2.1 Set theory
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2.1.1 Basic Terminology 2.1.2 Set Operations 2.1.3 Sets of Sets Set Definition and Equality Definition: A set is a collection of well-defined objects, called elements, that usually share some common attributes, but are not otherwise restricted. Many ways to specify the contents of a set: Listing elements explicitly: A = { 1 , 2 , 3 , 4 , 5 , 6 } Stating common property: A = { a : a positive int. 6 } If a is an element of set A, we write a A ; if a is not an element of A , we write a n∈ A . Definition: Two sets A and B are identical (or equal ) if and only if (iff) they have the same elements, in which case we write A = B . If A and B are not identical, we write A n = B . Example: A = { 1 , 2 , . . . , 6 } and B = { 2 , 4 , 6 } ⇒ A n = B ECSE 305 2.1 Set theory
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2.1.1 Basic Terminology 2.1.2 Set Operations 2.1.3 Sets of Sets Subsets Definition: A set A is a subset of a set B ( A is contained in B ) iff every element of the set A is also an element of B . Definition: A set A is a proper subset of set B iff A is a subset of B; AND there exists b B such that b n∈ A . Notation : A is a subset of B A B A is not a subset of B A n⊆ B A is a proper subset of B A B A is not a proper subset of B A n⊂ B Example: Let A = { 1 , 2 , ..., 6 } , B = { 2 , 4 , 6 } and C = { 0 , 1 } , then B A , B A , C n⊆ A , etc. ECSE 305 2.1 Set theory
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2.1.1 Basic Terminology 2.1.2 Set Operations 2.1.3 Sets of Sets Sample Space and Empty Set Definition: All sets of interest are usually subsets of larger set, called sample space ( universal set ) and denoted S . Definition: The the set with no elements is called empty (or null ) and is denoted by . By definition: s S , s / ∈ ∅ Theorem: Let A , B , and C denote arbitrary subsets of a sample space S . The following relations hold: (a) A A (b) A B and B C implies A C (c) A = B if and only if A B and B A (d) ∅ ⊆ A S ECSE 305 2.1 Set theory
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2.1.1 Basic Terminology 2.1.2 Set Operations 2.1.3 Sets of Sets Countable vs Uncountable Set A set is called finite if it is empty or contains a finite number of elements; otherwise it is called infinite . A set is called countable if is finite ( countably finite ), or if it is infinite but can be put into a one-to-one correspondence with the set of positive integers N ( countably infinite ). Otherwise the set is called uncountable . Examples: Countably finite: { 1 , 2 , 3 , 4 , 5 , 6 } . Countably infinite: N , Z , Q . Uncountably infinite: R , C , the open interval of the real line ( a , b ) = { x R : a < x < b } with a < b .
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This note was uploaded on 01/27/2011 for the course ECSE 305 taught by Professor Champagne during the Spring '09 term at McGill.

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Chapter 02 - 2.1.1 Basic Terminology 2.1.2 Set Operations...

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