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Unformatted text preview: Notes: c circlecopyrt F.P. Greenleaf, 20002010 v43f10sets.tex (version 9/01/10) Algebra I Chapter 1. Basic Facts from Set Theory 1.1 Glossary of abbreviations. Below we list some standard math symbols that will be used as shorthand abbreviations throughout this course and in the handout notes. • ∀ means “for all; for every” • ∃ means “there exists (at least one)” • ∃ ! means “there exists exactly one” • s.t. means “such that” • = ⇒ means “implies” • ⇐⇒ means “if and only if” • x ∈ A means “the point x belongs to a set A ” • N denotes the set of natural numbers (counting numbers) 1 , 2 , 3 , ··· • Z denotes the set of all integers (positive, negative or zero) • Q denotes the set of rational numbers • R denotes the set of real numbers • C denotes the set of complex numbers • { x ∈ A : P ( x ) } If A is a set, this denotes the subset of elements x in A such that statement P ( x ) is true. As examples of the last notation for specifying subsets: { x ∈ R : x 2 + 1 ≥ 2 } = ( −∞ , − 1] ∪ [1 , ∞ ) { x ∈ R : x 2 + 1 = 0 } = ∅ { z ∈ C : z 2 + 1 = 0 } = { + i, − i } where i = √ − 1 1.2 Basic facts from set theory. Next we review the basic definitions and notations of set theory, which will be used throughout our discussions of algebra. • ∅ denotes the empty set , the set with nothing in it • x ∈ A means that the point x belongs to a set A, or that x is an element of A . • A ⊆ B means A is a subset of B – i.e. any element of A also belongs to B (in symbolic notation: x ∈ A ⇒ x ∈ B ). The symbols A ⊆ B and B ⊇ A are used interchangeably. • A = B means the sets A and B contain exactly the same points. This statement is equivalent to saying: A ⊆ B and B ⊆ A . 1 • If a set consists of just one point p it is called a singleton set , denoted { p } . (Logically speaking, the “point p ” is not the same thing as “the set { p } whose only element is p ,” which is why we need a distinctive notation for singletons.) • A ∩ B indicates the intersection of two sets. An element x is in A ∩ B ⇔ x ∈ A and x ∈ B . Notice that A ∩ B = B ∩ A . • A ∪ B indicates the union of two sets. An element x lies in A ∪ B ⇔ either x ∈ A or x ∈ B ( or both ). Notice that A ∪ B = B ∪ A . Intersections and unions of several sets A 1 ,...,A n are indicated by writing n intersectiondisplay i =1 A i = A 1 ∩ ... ∩ A n This is the set { x : x ∈ A i for every i } n uniondisplay i =1 A i = A 1 ∪ ... ∪ A n This is the set { x : ∃ some i such that x ∈ A i } However, this notation is not practical when we wish to discuss unions or intersections of huge collections of sets. To handle those we use the following notation: Let I by a set of indices and suppose that we have assigned a set A α to each index α ∈ I . Then the intersection and union of the sets in this collection are denoted intersectiondisplay α ∈ I A α...
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 Spring '11
 ALgebra
 Algebra, Set Theory, Sets, Equivalence relation, Cartesian product, equivalence classes, Basic concepts in set theory, rst relation

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