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Unformatted text preview: Lecture 14: Functions June 21, 2010 A set is a collection of things. All mathematical objects are sets. Usually, they have some additional structure. One kind of structure that we find in many of the sets that occur in mathematics is the presence of operations , such as + or . Another kind of structure is the relation of order . Set theory is a branch of mathematics that was invented somewhat over 100 years ago. The purpose of set theory is to study the properties of sets apart from any additional structure. Two sets are equivalent (from the point of view of set theory) if they can be put into a one-to-one correspondence, with every element of the first set corresponding with exactly one element of the second, and every element of the second set receiving correspondence from exactly one element of the first. One of the first great observations of set theory was that the set of rational numbers Q is equivalent to the set of integers Z , but not to the set of real numbers Z .....
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