{"id":35720,"date":"2018-02-20T23:50:30","date_gmt":"2018-02-20T23:50:30","guid":{"rendered":"https:\/\/writemyessayfree.com\/the-set-of-integers-with-the-operation"},"modified":"2017-08-16T06:46:52","modified_gmt":"2017-08-16T06:46:52","slug":"the-set-of-integers-with-the-operation","status":"publish","type":"post","link":"https:\/\/www.benedictsol.com\/blogs\/the-set-of-integers-with-the-operation\/","title":{"rendered":"The set of integers with the operation ("},"content":{"rendered":"<div class=\"the_content_wrapper\">\n<p>Math 429 Test #1 Provide all work in the space provided! Name___________________<br \/> 1. The set of integers with the operation ( ) 11 a b a b * = + \u2013 , where ,a b Z ? , satisfies the conditions needed to define a group. Find and verify the identity element and the inverse of aZ ? ,with respect to this operation. Is it an abelian group?<br \/> 2. Using this exact wording and only this one sentence, finish the following. a) If f is surjective and :f C D ? , then each element of \u2026..<br \/> b) If f is injective and :f C D ? , then each element of \u2026\u2026<br \/> 3. Define what we mean by isomorphic binary structures.<br \/> 4. Prove that the identity element of any group is unique.<br \/> 5. For the following mapping, determine if the mapping is one to one and if it is onto. Justify all your answers with algebraic evidence and\/or algebraically justify all counter-examples. 35 5 () x x fx \u2013 + = where : { 5} f R R -?<br \/> 6. For any group G, prove that for any 1 1 1 *, ,( * ) a b G a b b a \u2013 \u2013 \u2013 ?= .<br \/> 7. Determine whether the following binary operations give a group structure on the given set. If not, give a reason they fail and supply a counter-example. You do not need to prove or show algebraic evidence for the properties that are satisfied.<br \/> a) The set of all positive rational numbers with the operation of multiplication.<br \/> b) The set of all even integers with the operation of multiplication.<br \/> c) The rational numbers under multiplication.<br \/> d) The set of all complex numbers z that have an absolute value of 1 with multiplication.<br \/> e) The set of all complex numbers z that have an absolute value of 1 with addition.<br \/> f) For a fixed positive integer n, the set of all complex numbers 1 nx = with multiplication.<br \/> 8. In the following, a relation R is defined on set of all human beings. Determine whether or not each relation is reflexive, symmetric, and\/or transitive.<br \/> a) xRy if and only if x lives within 400 miles of y.<br \/> b) xRy if and only if x is the father of y.<br \/> c) xRy if and only if x is the first cousin of y.<br \/> 9. Suppose Ga = is a cyclic group of order 12. List all distinct generators of G and list all distinct subgroups of G.<br \/> 10. Determine whether the following mappings are isomorphisms of the first binary structure with the second. If not, explain why and give a counter-example. a) 2 , , ( ) Q with Q by x x ? \u2022\u2022= for xQ ?<br \/> b) , , ( ) 2 xR with R by x ? + + \u2022 = for xR ?<br \/> 11. Find the order of the cyclic subgroup 8 U generated by 33 cos sin 22 i pp + . Show work.<br \/> 12. Let { } { } 2 : , ( ) , 2, 1,0,1,2 , 2,4,7,11 f Z Z such that f x x S and T ? = = \u2013 \u2013 = . If they exist, find the following.<br \/> a) () fS=<br \/> b) 1() fT \u2013 =<br \/> 13. Give the definition of a function and clearly specify what characteristics would allow it to be injective and\/or surjective.<br \/> 14. Give a complete definition of a group. <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Math 429 Test #1 Provide all work in the space provided! Name___________________ 1. The set of integers with the operation ( ) 11 a b a b * = + \u2013 , where ,a b Z ? , satisfies the <a href=\"https:\/\/www.benedictsol.com\/blogs\/the-set-of-integers-with-the-operation\/\" class=\"read-more\">Read More &#8230;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[],"tags":[],"class_list":["post-35720","post","type-post","status-publish","format-standard","hentry"],"_links":{"self":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/posts\/35720","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/comments?post=35720"}],"version-history":[{"count":0,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/posts\/35720\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/media?parent=35720"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/categories?post=35720"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.benedictsol.com\/blogs\/wp-json\/wp\/v2\/tags?post=35720"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}