Anki Deck Changes

Note 1: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: N9}Teh]+={

modified

Before

Front

For $H$ to be a subgroup, it must have closure under inverses: {{c2:: $\widehat{a} \in H$ for all $a \in H$}}.

Back

For $H$ to be a subgroup, it must have closure under inverses: {{c2:: $\widehat{a} \in H$ for all $a \in H$}}.

After

Front

For $H$ to be a subgroup, it must have closure under inverses: {{c1:: $\widehat{a} \in H$ for all $a \in H$}}.

Back

For $H$ to be a subgroup, it must have closure under inverses: {{c1:: $\widehat{a} \in H$ for all $a \in H$}}.

Field-by-field Comparison

Field	Before	After
Text	<p>For $H$ to be a subgroup, it must have {{c1::closure under inverses}}: {{c2:: $\widehat{a} \in H$ for all $~~{{c3::~~a \in H}}$}}.</p>	<p>For $H$ to be a subgroup, it must have {{c1::closure under inverses}}: {{c1:: $\widehat{a} \in H$ for all $a \in H$}}.</p>

Tags: ETH::1._Semester::DiskMat::5._Algebra::3._The_Structure_of_Groups::3._Subgroups

Note 2: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: cAat^jY(>E

modified

Before

Front

The set of units of $R$ is denoted by $R^*$ and $R^*$ is a group. This holds as we can easily see that every element of $R^*$ has an inverse by definition. Thus the axiom $G3$ holds.

Back

The set of units of $R$ is denoted by $R^*$ and $R^*$ is a group. This holds as we can easily see that every element of $R^*$ has an inverse by definition. Thus the axiom $G3$ holds.

After

Front

The set of units of $R$ is denoted by $R^*$ and it is a group. This holds as we can easily see that every element of $R^*$ has an inverse by definition. Thus the axiom $G3$ holds.

Back

The set of units of $R$ is denoted by $R^*$ and it is a group. This holds as we can easily see that every element of $R^*$ has an inverse by definition. Thus the axiom $G3$ holds.

Field-by-field Comparison

Field	Before	After
Text	<p>The {{c1::set of units}} of $R$ is denoted by {{c2::$R^$}} and ~~{{c3::$R^$~~ is a group. This holds as we can easily see that every element of $R^*$ has an inverse by definition. Thus the axiom $G3$ holds}}.</p>	<p>The {{c1::set of units}} of $R$ is denoted by {{c1::$R^$}} and it {{c3::is a group. This holds as we can easily see that every element of $R^$ has an inverse by definition. Thus the axiom $G3$ holds}}.</p>

Tags: ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::2._Units_and_the_Multiplicative_Group_of_a_Ring

Note 3: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: iltVkN7$2X

modified

Before

Front

A right inverse element of $a$ in $\langle S; *, e \rangle$ is an element $b$ such that $a * b = e$.

Back

A right inverse element of $a$ in $\langle S; *, e \rangle$ is an element $b$ such that $a * b = e$.

After

Front

A right inverse element of $a$ in $\langle S; *, e \rangle$ is an element $b$ such that $a * b = e$.

Back

A right inverse element of $a$ in $\langle S; *, e \rangle$ is an element $b$ such that $a * b = e$.

Field-by-field Comparison

Field	Before	After
Text	<p>A {{c1::right inverse element}} of $a$ in $\langle S; , e \rangle$ is an element $b$ such that ~~{{c3::~~$a b = e$}}.</p>	<p>A {{c1::right inverse element}} of $a$ in $\langle S; , e \rangle$ is {{c3::an element $b$ such that $a b = e$}}.</p>

Tags: ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups

Note 4: ETH::DiskMat

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

oFF!4


            modified
            

            
            
            
                
                    Before
                    
                    
                        Front
                        
                            
  ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
  
  
    A group \(\langle G; * \rangle\) (or monoid) is called commutative or abelian if \(a * b = b * a\) for all \(a, b \in G\).
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
  
  
    A group \(\langle G; * \rangle\) (or monoid) is called commutative or abelian if \(a * b = b * a\) for all \(a, b \in G\).
    
    
  



                        
                    
                    
                
                
                    After
                    
                    
                        Front
                        
                            
  ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
  
  
    A group \(\langle G; * \rangle\) (or monoid) is called commutative or abelian if \(a * b = b * a\) for all \(a, b \in G\).
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
  
  
    A group \(\langle G; * \rangle\) (or monoid) is called commutative or abelian if \(a * b = b * a\) for all \(a, b \in G\).
    
    
  



                        
                    
                    
                
            
            

            
            
            
                Field-by-field Comparison
                
                    
                        
                            Field
                            Before
                            After
                        
                    
                    
                        
                        
                            Text
                            <p>A group \(\langle G; * \rangle\) (or monoid) is called {{c1::commutative}} or {{c1::abelian}} if {{c2::\(a * b = b * a\)}} for all \({{c3::a, b}} \in G\).</p>
                            <p>A group \(\langle G; * \rangle\) (or monoid) is called {{c1::commutative}} or {{c1::abelian}} if {{c2::\(a * b = b * a\)}} for all \(a, b \in G\).</p>
                        
                        
                    
                
            
            

            
            
            
                Tags:
                
                    
                    
                    
                    
                    ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups

Field	Before	After
Text	<p>A group \(\langle G; * \rangle\) (or monoid) is called {{c1::commutative}} or {{c1::abelian}} if {{c2::\(a * b = b * a\)}} for all \(~~{{c3::~~a, b}} \in G\).</p>	<p>A group \(\langle G; * \rangle\) (or monoid) is called {{c1::commutative}} or {{c1::abelian}} if {{c2::\(a * b = b * a\)}} for all \(a, b \in G\).</p>


        
        
            Note 5: ETH::DiskMat
            
                Deck: ETH::DiskMat

                Note Type: Horvath Cloze

                GUID: p9^,`U1Fb;
            
            modified
            

            
            
            
                
                    Before
                    
                    
                        Front
                        
                            
  ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::5._Polynomial_Rings
  
  
    The degree of the product of two polynomials is equal to the sum of their degrees if \(R\) is an integral domain.
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::5._Polynomial_Rings
  
  
    The degree of the product of two polynomials is equal to the sum of their degrees if \(R\) is an integral domain.
    
    
  



                        
                    
                    
                
                
                    After
                    
                    
                        Front
                        
                            
  ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::5._Polynomial_Rings
  
  
    The degree of the product of two polynomials is equal to the sum of their degrees if \(R\) is an integral domain.
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::5._Polynomial_Rings
  
  
    The degree of the product of two polynomials is equal to the sum of their degrees if \(R\) is an integral domain.
    
    
  



                        
                    
                    
                
            
            

            
            
            
                Field-by-field Comparison
                
                    
                        
                            Field
                            Before
                            After
                        
                    
                    
                        
                        
                            Text
                            <p>The degree of the product of two polynomials is {{c1::equal}} to the sum of their degrees if \(R\) is an {{c2::integral domain}}.</p>
                            <p>The degree of the product of two polynomials is {{c1::equal to the sum of their degrees}} if \(R\) is an {{c2::integral domain}}.</p>
                        
                        
                    
                
            
            

            
            
            
                Tags:
                
                    
                    
                    
                    
                    ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::5._Polynomial_Rings