By Frederick M. Goodman
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In trouble-free introductions to mathematical research, the remedy of the logical and algebraic foundations of the topic is unavoidably particularly skeletal. This publication makes an attempt to flesh out the bones of such remedy through offering a casual yet systematic account of the rules of mathematical research written at an user-friendly point.
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Extra resources for Algebra: Abstract and Concrete (Stressing Symmetry) (2.5 Edition)
C) Divisibility is transitive: If ajb and bjc, then ajc. (d) If ajb and ajc, then a divides all integers that can be expressed in the form sb C t c, where s and t are integers. Proof. For part (a), note that neither u nor v can be zero. Suppose first that both are positive. We have uv maxfu; vg 1. If equality holds, then u D v D 1. In general, if uv D 1, then also jujjvj D 1, so juj D jvj D 1. Thus both u and v are ˙1. Since their product is positive, both have the same sign. For (b), let u; v be integers such that b D ua and a D vb.
The first of the permutations takes 1 to 4 and the second takes 4 to 7, so the product takes 1 to 7. The first leaves 7 fixed and the second takes 7 to 6, so the product takes 7 to 6. The first takes 6 to 5 and the second takes 5 to 4, so the product takes 6 to 4. The first takes 4 to 2 and the second leaves 2 fixed, so the product takes 4 to 2. The first takes 2 to 3 and the second takes 3 to 1, so the product takes 2 to 1. 1 7 6 4 2/. The first permutation takes 5 to 6 and the second takes 6 to 5, so the product fixes 5.
Mod n/. Proof. a C b/ a0 and b b 0 are divisible by n. a is divisible by n. b a0 b 0 / b0/ ■ 40 1. ALGEBRAIC THEMES We denote by Zn the set of residue classes modulo n. The set Zn has a natural algebraic structure which we now describe. Let A and B be elements of Zn , and let a 2 A and b 2 B; we say that a is a representative of the residue class A, and b a representative of the residue class B. The class Œa C b and the class Œab are independent of the choice of representatives. 5. Thus Œa C b D Œa0 C b 0 and Œab D Œa0 b 0 .
Algebra: Abstract and Concrete (Stressing Symmetry) (2.5 Edition) by Frederick M. Goodman