By Hwang A.

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0) (∃δ > 0) |x − x0 | < δ =⇒ |f (x) − f (x0 )| < ε. The next exercise introduces some basic properties of quantifiers. 7 Consider the quantified sentences: (a) For every marble x in the bag, x is blue. 4. CALCULUS AND THE “REAL WORLD” 31 (b) There exists a red marble y in the bag. (c) For every real x > 0, there exists a natural number n with 1/n < x. Give the negation of each sentence. Express (a) and (c) as conditional statements, and give their contrapositives. Express the negations of (b) and (c) as conditional statements, and give their contrapositives.

Roughly, the relation sees only whether x and y are related or not, and does not otherwise distinguish pairs of elements. 48 CHAPTER 2. 6: Equality. 7: Inequality. m • (Transitivity) For all x, y, and z ∈ X, if x ∼ y and y ∼ z, then x ∼ z. Intuitively, the relation is all-encompassing; everything related to something related to x is itself related to x. If ∼ is an equivalence relation on a set X, then there is a “partition” of X into disjoint subsets called equivalence classes, each consisting of elements that are mutually related by ∼.

There is a fringe benefit: The techniques of recursive definition, mathematical induction, and equivalence classes, which arise naturally in constructing the integers from set theory, are important and useful throughout mathematics. By far the most complicated step is the definition of real numbers in terms of rational numbers. If the recursive definitions are expanded, a single real number is a mind-bogglingly complicated set. Luckily, it is never necessary to work with “expanded” definitions; the abstract properties satisfied by the set of real numbers are perfectly adequate.