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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
onetoone 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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 Fall '11
 Wong
 Integers, Sets

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