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Math 546Homework 1 Due Wednesday, January 25. This homework has two types of problems. ã  546 Problems.  All students (students enrolled in 546  and   701I) are required to turn these in. ã  701I Problems.  Only students enrolled in 701I are required to turn these in. Students notenrolled in 701I are welcome to turn these in as well. I especially welcome students lookingfor a challenge to attempt these. Note:  Part 3 below contains solutions to some of the  unassigned   problems from Sections 1 and 2of Saracino’s text. I am providing these as examples of how to write solutions. Please have a look at these, and make sure to  read Sections 1 and 2  of the text. 1 546 Problems. 1.3and1.6.  Foreachofthefollowingsets S   andfunctions ∗ on S  × S  , determinewhether ∗ isa binary operation on  S  . If  ∗ is a binary operation on  S  , determine whether it is commutativeand whether it is associative. a)  S   = Z ,  a ∗ b  =  a  +  b 2 . e)  S   = Z ,  a ∗ b  =  a  +  b − ab . f)  S   = R ,  a ∗ b  =  b . g)  S   =  { 1 , − 2 , 3 , 2 , − 4 } ,  a ∗ b  =  | b | . 1.4.  Is division a commutative operation on R + ? Is it associative? Note:  If   S   ⊆ R , then we deﬁne  S  + =  { s  ∈  S   :  s >  0 } . 1.9.  Let  S   =  { a,b,c,d } . The following table deﬁnes a binary operation ∗ on  S  .Is ∗ commutative? Is it associative?* a b c da a c b db c a d bc b d a cd d b c a Note:  For  s 1 ,  s 2  ∈  S  , we deﬁne  s 1  ∗ s 2  to the be the element in the row that contains  s 1  andthe column that contains  s 2 . For example, we have  c ∗ b  =  d .  2.1.  Which of the following are groups? Why or why not? d)  The set { 1 , − 1 } under multiplication. f)  R × R =  { ( x,y ) :  x,y  ∈ R } under the operation  ( x,y ) ∗ ( z,w ) = ( x  +  z,y  − w ) . g)  R × R × =  { ( x,y ) :  x,y  ∈ R and  y   = 0 } under the operation ( x,y ) ∗ ( z,w ) = ( x  +  z,yw ) . Recall that R × = R \{ 0 } . h)  R \{ 1 } under the operation  a ∗ b  =  a  +  b − ab . 2.5.  Let  S   =  { a,b,c } . The following table deﬁnes a binary operation ∗ on  S  .* a b ca a b cb b b cc c c cIs  ( S, ∗ )  a group? Why or why not? 2.8.  Let  G  =  { f   : R → R :  for all  x  ∈ R , f  ( x )   = 0 } . For all  f  ,  g  ∈  G , deﬁne × by ( f   × g )( x ) =  f  ( x ) g ( x )  for all  x  ∈ R . Is  ( G, × )  a group? Prove or disprove. 2.11.  Let  G  =  a  00  b  :  a,b   = 0  in R  . Show that  G  forms a group under matrix multi-plication. 2 701I Problems. 1.10.  How many binary operations are there on a set  S   with  n  elements? How many of theseare commutative? 2.1 e)  Let  S   =  { q   ∈ Q + :  √  q   ∈ Q } ,  ∗ is × . Is  ( S, ∗ )  a group? Why or why not?  2.6.  Let  S   =  { a,b,c } . The following table deﬁnes a binary operation ∗ on S.* a b ca a b cb b a cc c b aIs  ( S, ∗ )  a group? Why or why not? 2.10.  Let G  =  M   =   a b − b a  :  a,b  ∈ R ,  det  M   =  a 2 +  b 2  = 0  ⊂  GL (2 , R ) . Let × denote ordinary matrix multiplication. Show that  ( G, × )  is a group. 3 Examples. 1.3and1.6.  Foreachofthefollowingsets S   andfunctions ∗ on S  × S  , determinewhether ∗ isa binary operation on  S  . If  ∗ is a binary operation on  S  , determine whether it is commutativeand whether it is associative. b)  S   = Z , a ∗ b  =  a ∗ b  =  a 2 b 3 . Claim:  ∗ is a binary operation on  S  . Proof:  For all  a ,  b  ∈ Z , we have  a ∗ b  =  a 2 b 3 ∈ Z , so ∗ is a binary operation. Claim:  ∗ is not commutative on  S  . Proof:  We note that  b  ∗  a  =  b 2 a 3 . Therefore, we have  a  ∗  b  =  b  ∗  a  if and only if  a 2 b 3 =  a 3 b 2 , which holds if and only if   a  =  b  or either of   a ,  b  = 0 . We observe that 1 ∗ 2 = 1 2 2 3 = 8   = 4 = 2 2 1 3 = 2 ∗ 1 . Hence, ∗ is not commutative. Claim:  ∗ is not associative on  S  . Proof:  We compute ( a ∗ b ) ∗ c  = ( a 2 b 3 ) ∗ c  = ( a 2 b 3 ) 2 c 3 =  a 4 b 6 c 3 ,a ∗ ( b ∗ c ) =  a ∗ ( b 2 c 3 ) =  a 2 ( b 2 c 3 ) 3 =  a 2 b 6 c 9 . Therefore, we have  ( a ∗ b ) ∗ c  =  a ∗ ( b ∗ c )  if and only if   a 4 b 6 c 3 =  a 2 b 6 c 9 , which holdsfor  a ,  b ,  c   = 0  if only if   a 2 =  c 6 . Taking square roots, we see that associativity followsif and only if   a  =  c 3 . We note that (2 ∗ 1) ∗ 1 = 2 4 1 6 1 3 = 16   = 4 = 2 2 1 6 1 9 = 2 ∗ (1 ∗ 1) . Hence, ∗ is not associative.  h)  S   =  { 1 , 6 , 3 , 2 , 18 } , a ∗ b  =  ab . Claim:  ∗ is not a binary operation on  S  . Proof:  We note that  6 ,  2  ∈  S  , but that  6  ∗  2 = 12  ∈  S  . Therefore,  ∗  is not a binaryoperation.Since  ∗  is not a binary operation, we do not consider whether or not  ∗  is commutativeor associative. 2.1.  Which of the following are groups? Why or why not? b)  S   = 3 Z =  { 3 n  :  n  ∈ Z } , the set of integers that are multiples of   3 ,  ∗ is  + . Claim:  ( S, ∗ )  is a group. Proof:  It sufﬁces to verify the axioms:(i)  ∗ is a binary operation on  S  . To see this, let  a  = 3 x ,  b  = 3 y  ∈  3 Z . Then we have a  +  b  = 3 x  + 3 y  = 3( x  +  y )  ∈  3 Z , so ∗ is a binary operation on  S  .(ii)  ∗  is associative on  S  . Since  S   = 3 Z  ⊂  Z , we see that  3 Z  inherits associativityunder  +  from Z .(iii)  ( S, ∗ )  has an identity. We claim that  0 = 3 · 0  ∈  3 Z is an identity for  ( S, ∗ ) . To seethis, let  a  = 3 x  ∈  3 Z . Then we have a  + 0 = 3 x  + 0 = 3 x  = 0 + 3 x  = 0 +  a, so  0  is an identity for  ( S, ∗ ) .(iv)  ( S, ∗ )  has inverses. To see this, let  a  = 3 x  ∈  3 Z . Then − a  =  − 3 x  = 3( − x )  ∈  3 Z has − a  +  a  = 0 =  a  + ( − a ) , so − a  is an inverse for  a .It follows that  ( S, ∗ )  is a group. Furthermore,  ( S, ∗ )  inherits commutativity from  Z since  S   ⊆ Z , so  ( S, ∗ )  is an abelian group. i)  S   = Z , a ∗ b  =  a  +  b − 1 . Claim:  ( S, ∗ )  is a group. Proof:  It sufﬁces to verify the axioms.(i)  ∗ is a binary operation on  S  . To see this, let  a ,  b  ∈ Z . Then we have a ∗ b  =  a  +  b − 1  ∈ Z , so ∗ is a binary operation on  S  .(ii)  ∗ is associative on  S  . To see this, let  a ,  b ,  c  ∈ Z . Then we have ( a ∗ b ) ∗ c  = ( a  +  b − 1) ∗ c  = ( a  +  b − 1) +  c − 1 =  a  + ( b  +  c − 1) − 1=  a ∗ ( b  +  c − 1) =  a ∗ ( b ∗ c ) , so ∗ is associative on  S  .

Sep 10, 2019

#### BC First Part

Sep 10, 2019
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