Some Notes on Sets, Logic, and Mathematical Language

Some Notes on Sets, Logic, and Mathematical Language -...

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Unformatted text preview: George M. Bergman Spring 2006 Supplementary material Some notes on sets, logic, and mathematical language These are generic notes, for use in Math 110, 113, 104 or 185. These pages do not develop in detail the de f nitions and concepts to be mentioned. That is done, to various degrees, in Math 55, Math 74, Math 125 and Math 135. I hope you will nevertheless f nd them useful and thought-provoking. I recommend working the exercises for practice; but dont hand them in unless they are listed in a homework assignment for the course. 1. Set-theoretic symbols Symbol. Meaning, usage, examples, discussion. ! The empty set. , Here denotes the set of all natural numbers , i.e., { 0, 1, 2, 3,... } , while (from Zahl, German for number) denotes the set of all integers, { ...,3, 2, 1, 0, 1, 2, 3,... } . (Many older authors started the natural numbers with 1, but it is preferable to start with 0, since natural numbers are used to count the elements of f nite sets, and ! is a f nite set.) , , Of these, (for quotient) denotes the set of all rational numbers , i.e., fractions that can be written with integer numerator and denominator, denotes the set of real numbers , and the set of complex numbers . (The f ve sets just named used to be, and often still are, denoted by bold-face letters N , Z , Q , R and C . The forms , ..., arose as quick ways to write these boldface letters on the blackboard. Since it is convenient to have distinctive symbols for these important sets, printed forms imitating the blackboard bold symbols were then designed, and are now frequently used, as shown.) " Is a member of. E.g., 3 " . { } The set of all. This is often used together with or : (different authors prefer the one or the other), which stand for such that. For instance, the set of positive integers can be written { 1,2,3,... } or { n " n >0 } or { n : n " , n >0 } . The set of all square integers can be written { 0,1,4,9,..., n 2 ,... } or { n 2 n " } . Note also that { n 2 n " } = { m 2 m " } = { m 2 +2 m +1 m " } . (Why?) # Is a subset of. E.g., ! # { 1 } # # # # . { n 2 n " } # { n " n $ } . # . % or % & Is a proper subset of; that is, a subset that is not the whole set. For instance, % & . In fact, all the formulas used above to illustrate # remain true with % & in place of # except for # . Since a proper subset is, in particular, a subset, one may use # even when % & is true; and one generally does so, unless one wants to emphasize that a subset is proper. But beware: some authors (especially in Eastern Europe) use % for is a subset of....
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This note was uploaded on 05/04/2011 for the course MATH 73 taught by Professor Danberwick during the Spring '09 term at University of California, Berkeley.

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Some Notes on Sets, Logic, and Mathematical Language -...

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