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Talking about Sets

Talking about Sets - Math 100 Talking about sets Martin H...

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Math 100 Talking about sets Martin H. Weissman We have been building up the language of mathematics, and this continues today. Here are two kinds of sentences that we have discussed so far: Sentences which express facts about quantities. For example, the sentences: (1) 3 + x > 12 . (2) x Z such that x 2 = 16 . (3) x R , x 2 + 1 x. Sentences which name or choose a variable: (1) Let x be the square root of π . (2) Let y be the height of the Sears tower (in feet). (3) Let z be x + 3. (4) Let n be an odd number. Even with these basic kinds of sentences, we can express some pretty advanced mathematical facts. Here are some examples of facts which we can express: Every positive integer can be expressed as the sum of four perfect squares. (Legendre’s Theorem) If x, y, z are integers, and x 3 + y 3 = z 3 , then xyz = 0. (A result of Euler, following Fermat). To increase the expressive power of mathematics even further, it is necessary to talk about sets. 1. The basic language of sets We have already used a bit of the language of sets, when discussing quantifiers. A “quantifying phrase”, such as x R , or y Z , involves a quantifier (either or ), a variable (such as x or y ), the symbol “ ”, and a set , such as R , Z , Q , C , or N . In practice, we would like to use many more sets than these five. There are many ways of defining a set, the simplest of which is... 1.1. Directly defining a set. One may directly define a finite set with a sentence like: “Let S = { 1 , 3 , - 7 , π } ”, where the “stuff” between the braces consists of a list (separated by commas) of numbers or even other sets. Only use this direct definition of a set when defining a finite set. Some people use such notation when defining infinite sets, with a sentence like “Let S = { 2 , 3 , 4 , 5 , ... } ”, but this leads to practical and philosophical problems.
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