Anki Deck Changes

Note 1: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: c.BJE1FC)A

modified

Before

Front

A binary operation $*$ on a set $S$ is associative if $a * (b * c) = (a * b) * c$ for all $a, b, c$ in $S$.

Back

A binary operation $*$ on a set $S$ is associative if $a * (b * c) = (a * b) * c$ for all $a, b, c$ in $S$.

After

Front

A binary operation $*$ on a set $S$ is associative if $a * (b * c) = (a * b) * c$ for all $a, b, c$ in $S$.

Back

A binary operation $*$ on a set $S$ is associative if $a * (b * c) = (a * b) * c$ for all $a, b, c$ in $S$.

Field-by-field Comparison

Field	Before	After
Text	<p>A binary operation $$ on a set $S$ is {{c1::associative}} if {{c2::$a (b * c) = (a * b) * c$}} for all $~~{{c3::~~a, b, c}}$ in $S$.</p>	<p>A binary operation $$ on a set $S$ is {{c1::associative}} if {{c2::$a (b * c) = (a * b) * c$}} for all $a, b, c$ in $S$.</p>

Tags: ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::2._Associativity_and_Monoids

Note 2: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: grVf##]DMH

modified

Before

Front

Lemma 5.17(4): If a ring $R$ is non-trivial (has more than one element), then $1 \neq 0$.

Back

Lemma 5.17(4): If a ring $R$ is non-trivial (has more than one element), then $1 \neq 0$.

After

Front

Lemma 5.17(4): If a ring $R$ is non-trivial (has more than one element), then $1 \neq 0$.

Back

Lemma 5.17(4): If a ring $R$ is non-trivial (has more than one element), then $1 \neq 0$.

Proof: Assume $1 = 0$ for contradiction. For any $a \in R$

$a = a \cdot 1$
$a = a \cdot 0$ (by assumption)
$a = 0$
Thus there is only the zero element, which is a contradiction to the non-triviality.

Field-by-field Comparison

Field	Before	After
Extra		Proof: Assume $1 = 0$ for contradiction. For any $a \in R$<br><ol><li>$a = a \cdot 1$</li><li>$a = a \cdot 0$ (by assumption)</li><li>$a = 0$</li><li>Thus there is only the zero element, which is a contradiction to the non-triviality.</li></ol>

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

Note 3: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: oPaK;$.R2B

modified

Before

Front

An irreducible polynomial of degree $\geq 2$ has no roots in the field.

Proof: If it had a root $\alpha$, then $(x - \alpha)$ would divide it by Lemma 5.29, contradicting irreducibility.

Back

An irreducible polynomial of degree $\geq 2$ has no roots in the field.

Proof: If it had a root $\alpha$, then $(x - \alpha)$ would divide it by Lemma 5.29, contradicting irreducibility.

After

Front

An irreducible polynomial has no roots in the field. It has to have degree $\geq 2$.

Back

An irreducible polynomial has no roots in the field. It has to have degree $\geq 2$.

Proof: If it had a root $\alpha$, then $(x - \alpha)$ would divide it by Lemma 5.29, contradicting irreducibility.

Field-by-field Comparison

Field	Before	After
Text	<p>An {{c1::irreducible}} polynomial ~~of degree {{c2::$\geq 2$}}~~ has {{c3::no roots}} in the field.~~</p> <p><strong>Proof</strong>: If it had a root $\alpha$, then $(x - \alpha)$ would divide it by Lemma 5.29, contradicting irreducibility~~.</p>	<p>An {{c1::irreducible}} polynomial has {{c3::no roots}} in the field. It has to have {{c2::degree $\geq 2$}}.</p>
Extra		<strong>Proof</strong>: If it had a root $\alpha$, then $(x - \alpha)$ would divide it by Lemma 5.29, contradicting irreducibility.

Tags: ETH::1._Semester::DiskMat::5._Algebra::7._Polynomials_as_Functions::2._Roots

Note 4: ETH::DiskMat

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

s`dg


            modified
            

            
            
            
                
                    Before
                    
                    
                        Front
                        
                            
  ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
  
  
    If both \(b * a = e\) and \(a * b = e\), then \(b\) is simply called an inverse of \(a\).
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
  
  
    If both \(b * a = e\) and \(a * b = e\), then \(b\) is simply called an inverse of \(a\).
    
    
  



                        
                    
                    
                
                
                    After
                    
                    
                        Front
                        
                            
  ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
  
  
    If both \(b * a = e\) and \(a * b = e\), then \(b\) is simply called an inverse of \(a\).
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::DiskMat::5._Algebra::2._Monoids_and_Groups::3._Inverses_and_Groups
  
  
    If both \(b * a = e\) and \(a * b = e\), then \(b\) is simply called an inverse of \(a\).
    
    
  



                        
                    
                    
                
            
            

            
            
            
                Field-by-field Comparison
                
                    
                        
                            Field
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                            Text
                            <p>If both {{c1::\(b * a = e\)}} and {{c2::\(a * b = e\)}}, then \({{c3::b}}\) is simply called an {{c4::inverse}} of \(a\).</p>
                            <p>If both {{c1::\(b * a = e\)}} and {{c2::\(a * b = e\)}}, then \(b\) is {{c4::simply called an inverse of \(a\)}}.</p>
                        
                        
                    
                
            
            

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