Anki Deck Changes

Note 1: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: Q{f*UJkPmo

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Front

Relation and function composition is associative.

Back

Relation and function composition is associative.

This is important as \((\rho \circ \tau) \circ (\rho \circ \tau) = \rho \circ (\tau \circ \rho) \circ \tau\) is really useful in some exercises.

Field-by-field Comparison

Field	Before	After
Text		Relation and function composition is {{c1::associative}}.
Extra		This is important as \((\rho \circ \tau) \circ (\rho \circ \tau) = \rho \circ (\tau \circ \rho) \circ \tau\) is really useful in some exercises.

Tags: ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::3._Relations::5._Composition_of_Relations ETH::1._Semester::DiskMat::Exams::3._Relations::FS24

Note 2: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID:

jGU1:k0?


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  ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::5._Cyclic_Groups ETH::1._Semester::DiskMat::Exams::3._Algebra::FS23 Niklas_Skill_Issue
  
  
    Can we reduce \(R_9(5^{123})\) by doing \(R_9(123) = 6\)?
  



                        
                    
                    
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  ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::5._Cyclic_Groups ETH::1._Semester::DiskMat::Exams::3._Algebra::FS23 Niklas_Skill_Issue
  
  
    Can we reduce \(R_9(5^{123})\) by doing \(R_9(123) = 6\)?
    
    No! The order of \(5\) in \(\mathbb{Z}_9\) is \(\varphi(9) = 6\). Thus we reduce by \(R_6(123)\)!
  






                        
                    
                    
                
            

            

            
            
            
                Field-by-field Comparison
                
                    
                        
                            Field
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                            Front
                            
                            Can we reduce&nbsp;\(R_9(5^{123})\)&nbsp;by doing&nbsp;\(R_9(123) = 6\)?
                        
                        
                        
                            Back
                            
                            No! The order of&nbsp;\(5\)&nbsp;in&nbsp;\(\mathbb{Z}_9\)&nbsp;is&nbsp;\(\varphi(9) = 6\). Thus we reduce by&nbsp;\(R_6(123)\)!
                        
                        
                    
                
            
            

            
            
            
                Tags:
                
                    
                    
                    
                    
                    
                    
                    ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::5._Cyclic_Groups
                    
                    ETH::1._Semester::DiskMat::Exams::3._Algebra::FS23
                    
                    Niklas_Skill_Issue

Field	Before	After
Front		Can we reduce \(R_9(5^{123})\) by doing \(R_9(123) = 6\)?
Back		No! The order of \(5\) in \(\mathbb{Z}_9\) is \(\varphi(9) = 6\). Thus we reduce by \(R_6(123)\)!


        
        
            Note 3: ETH::DiskMat
            
                Deck: ETH::DiskMat

                Note Type: Horvath Cloze

                GUID: yAP=DE#~t<
            
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  ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups
  
  
    An abelian group has the following properties:
Closure
Associativity
Identity
Inverse
Commutativity
  



                        
                    
                    
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  ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups
  
  
    An abelian group has the following properties:
Closure
Associativity
Identity
Inverse
Commutativity
    
    
  



                        
                    
                    
                
            

            

            
            
            
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                            An <b>abelian group</b>&nbsp;has the following properties:<br><ol><li>{{c1::Closure}}</li><li>{{c2::Associativity}}</li><li>{{c3::Identity}}</li><li>{{c4::Inverse}}</li><li>{{c5::Commutativity}}</li></ol>
                        
                        
                    
                
            
            

            
            
            
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                    ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups
                    
                
            
            
        
        
        
            Note 4: ETH::DiskMat
            
                Deck: ETH::DiskMat

                Note Type: Horvath Cloze

                GUID: zT<$3-%Ev[
            
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  ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::5._Cyclic_Groups ETH::1._Semester::DiskMat::Exams::3._Algebra::FS23
  
  
    We can solve \(R_a(b^c)\) by using the fact that {{c1:: \(R_a(b^c) = R_a(b^{R_{\varphi(a)}(c)})\)}} if \(a, b\) coprime.
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::5._Cyclic_Groups ETH::1._Semester::DiskMat::Exams::3._Algebra::FS23
  
  
    We can solve \(R_a(b^c)\) by using the fact that {{c1:: \(R_a(b^c) = R_a(b^{R_{\varphi(a)}(c)})\)}} if \(a, b\) coprime.
    
    
    
    Note that we can't simply reduce by \(a\)!
  



                        
                    
                    
                
            

            

            
            
            
                Field-by-field Comparison
                
                    
                        
                            Field
                            Before
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                            Text
                            
                            We can solve&nbsp;\(R_a(b^c)\)&nbsp;by using the fact that {{c1::&nbsp;\(R_a(b^c) = R_a(b^{R_{\varphi(a)}(c)})\)}} if&nbsp;\(a, b\)&nbsp;coprime.
                        
                        
                        
                            Extra
                            
                            Note that we can't simply reduce by&nbsp;\(a\)!
                        
                        
                    
                
            
            

            
            
            
                Tags:
                
                    
                    
                    
                    
                    
                    
                    ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::5._Cyclic_Groups
                    
                    ETH::1._Semester::DiskMat::Exams::3._Algebra::FS23

Field	Before	After
Text		We can solve \(R_a(b^c)\) by using the fact that {{c1:: \(R_a(b^c) = R_a(b^{R_{\varphi(a)}(c)})\)}} if \(a, b\) coprime.
Extra		Note that we can't simply reduce by \(a\)!