Intro to Analysis in-class_Part_6

Intro to Analysis in-class_Part_6 - 1.1 Quantifiers 17...

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Unformatted text preview: 1.1 Quantifiers 17 Note: one implication is valid. ∃x, ∀y, A(x, y ) =⇒ ∀y, ∃x, A(x, y ). 1.1.3 Exercises: #3 Due: Jan. 29 Question 3. Interpret in words: ∀x, ∃y, y > x but not ∃y, ∀x, y > x. (x, y are integers) 18 Math 413 1.2 1.2.1 Preliminaries Infinite Sets Countable sets Definition 1.2.1. Two sets A and B have the same cardinality iff they can be put in one-to-one correspondence, i.e., if every element of A corresponds to a unique element of B ; all elements are “paired off”. Cardinality is “size” for finite sets, but it works for infinite sets, too. Definition 1.2.2. A set A is infinite iff there is a proper subset B ⊆ A which has the same cardinality as A. Definition 1.2.3. A set A is countable iff it has the same cardinality as the natural numbers N = {1, 2, 3, 4, . . . }, i.e., if we can write A = {a1 , a2 , a3 , . . . }. Are some infinite sets larger than others? Yes! Some sets have too many elements to count. Example 1.2.1. N is infinite (and countable): 1, 1, 2, 3, 4, . . . 2, 3, 4, 5, . . . Thus, a countable set is infinite. Theorem 1.2.4. The set of integers Z is countable. Proof. 0, 1, -1, 2, -2, . . . More precisely, define a function a : N → Z by an := n − , 2 n even, n+1 2 , n odd and convince yourself that it’s a bijection. ...
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Intro to Analysis in-class_Part_6 - 1.1 Quantifiers 17...

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