By A. I. Kostrikin, I. R. Shafarevich
This ebook is wholeheartedly prompt to each scholar or consumer of arithmetic. even if the writer modestly describes his ebook as 'merely an try to speak about' algebra, he succeeds in writing an exceptionally unique and hugely informative essay on algebra and its position in smooth arithmetic and technology. From the fields, commutative jewelry and teams studied in each collage math path, via Lie teams and algebras to cohomology and class thought, the writer exhibits how the origins of every algebraic suggestion should be relating to makes an attempt to version phenomena in physics or in different branches of arithmetic. related common with Hermann Weyl's evergreen essay The Classical Groups, Shafarevich's new booklet is certain to develop into required analyzing for mathematicians, from newbies to specialists.
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Extra info for Algebra 1. Basic Notions of Algebra
Is this clear? Regarding the first statement, integers are real numbers. Hence, the first statement is not true. However, the second statement is true because if you multiply any integer by 2 the result will still be an integer. Now, if we form the converse of each of these we obtain the following: 1. If x is not a real number, then x is an integer. 2. If 2x is an integer, then x is an integer. Note that the first of these statements is false (as was its converse), and the second is also false (as opposed to its converse which was true).
Hence, the first statement is not true. However, the second statement is true because if you multiply any integer by 2 the result will still be an integer. Now, if we form the converse of each of these we obtain the following: 1. If x is not a real number, then x is an integer. 2. If 2x is an integer, then x is an integer. Note that the first of these statements is false (as was its converse), and the second is also false (as opposed to its converse which was true). The point is that a statement and its converse are not logically equivalent and so knowing that a statement is true (or false) gives you no information as to whether or not its converse is true (or false).
This is because“10/5 = 2” states that “10 divided by 5 equals 2” which is a true statement, but “5|10 = 2” states that “5 divides 10 equals 2” which is not a statement. In fact, it’s not even a complete sentence. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience.