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.