By Joseph Gallian

Modern summary Algebra 7/e presents an excellent creation to the normal themes in summary algebra whereas conveying to scholars that it's a modern topic used day-by-day via operating mathematicians, laptop scientists, physicists, and chemists. The textual content comprises a variety of figures, tables, photos, charts, biographies, machine workouts, and instructed readings giving the topic a present consider which makes the content material attention-grabbing and correct for college kids.

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**Additional resources for Contemporary Abstract Algebra**

**Example text**

On the contrary, assume c [ [a] > [b]. We will show that [a] # [b]. To this end, let x [ [a]. We then have c , a, c , b, and x , a. By the symmetric property, we also have a , c. Thus, by transitivity, x , c, and by transitivity again, x , b. This proves [a] # [b]. Analogously, [b] # [a]. Thus, [a] 5 [b], in contradiction to our assumption that [a] and [b] are distinct equivalence classes. To prove the converse, let P be a collection of nonempty disjoint subsets of S whose union is S. Define a , b if a and b belong to the same subset in the collection.

EXAMPLE 10 Let S be the set of all triangles in a plane. If a, b [ S, define a , b if a and b are similarâ€”that is, if a and b have corresponding angles that are the same. Then, , is an equivalence relation on S. EXAMPLE 11 Let S be the set of all polynomials with real coefficients. If f, g [ S, define f , g if f 9 5 g9, where f 9 is the derivative of f. Then, , is an equivalence relation on S. Since two polynomials with equal derivatives differ by a constant, we see that for any f in S, [ f ] 5 { f 1 c | c is real}.

11. Let n be a fixed positive integer greater than 1. If a mod n 5 a9 and b mod n 5 b9, prove that (a 1 b) mod n 5 (a9 1 b9) mod n and (ab) mod n 5 (a9b9) mod n. ) 12. Let a and b be positive integers and let d 5 gcd(a, b) and m 5 lcm(a, b). If t divides both a and b, prove that t divides d. If s is a multiple of both a and b, prove that s is a multiple of m. 13. Let n and a be positive integers and let d 5 gcd(a, n). Show that the equation ax mod n 5 1 has a solution if and only if d 5 1. ) 14.