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Section1notes - Lecture 1 Introductory set theory the set of natural numbers and proof by induction 1 Basic set theory Denitions of subset set

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Lecture 1: Introductory set theory, the set of natural numbers and proof by induction. 1. Basic set theory Deﬁnitions of subset, set equality, union, intersection, disjointedness, empty set, power set, cartesian product. Ex: Prove that C \ ( A ¯ B ) = A ( ¯ B ¯ C ) . Note: Any statement “ x ∈ ∅ ...” is automatically true (i.e. a tautology). Set notation and indexing families. Ex: Let A k = [0 , 2 - 1 /k ]. Find S k =1 A k . Ex: Let B k = ( - 1 /k, 1 /k ). Find T k =1 B k . Deﬁnition of function, domain, range, image, pre-image. Ex: f ( f - 1 ( D )) D , f ( C 1 C 2 ) = f ( C 1 ) f ( C 2 ). What about ? Surjective, injective, and bijective functions. Deﬁnition of the inverse, and when it is a function. Ex: f ( x ) = e x , f ( x ) = x ( x - 1)( x + 1), f ( x ) = x 3 . Deﬁnitions of countable and uncountable. Ex: Z and Q are countable. Ex: R is uncountable. 2. The natural numbers and proof by induction Deﬁnition of the set
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This note was uploaded on 10/16/2011 for the course MATH 34 taught by Professor Wiley during the Winter '11 term at UC Merced.

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