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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
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 negationslash∈ 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 negationslash = B . Example: A = { 1 , 2 , . . . , 6 } and B = { 2 , 4 , 6 } ⇒ A negationslash = 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 negationslash∈ A . Notation : A is a subset of B A B A is not a subset of B A negationslash⊆ B A is a proper subset of B A B A is not a proper subset of B A negationslash⊂ B Example: Let A = { 1 , 2 , ..., 6 } , B = { 2 , 4 , 6 } and C = { 0 , 1 } , then B A , B A , C negationslash⊆ A , etc. ECSE 305 2.1 Set theory
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 .
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Chapter 02 - 2.1.1 Basic Terminology 2.1.2 Set Operations...

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