Kernel (set theory)

<p>In <a href="/facts/Set_theory/1ptfsvUP">set theory</a>, the kernel of a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 
  
    
      
        f
      
    
    {\displaystyle f}
  
 (or equivalence kernel) may be taken to be either
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<ul><li>the <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a> on the function's <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> that roughly expresses the idea of "equivalent as far as the function 
  
    
      
        f
      
    
    {\displaystyle f}
  
 can tell", or</li>
<li>the corresponding <a href="/facts/Partition_of_a_set/VeP9g8CA">partition</a> of the domain.</li></ul>
<p>An unrelated notion is that of the kernel of a non-empty <a href="/facts/Family_of_sets/WbYof7lP">family of sets</a> 
  
    
      
        
          
            B
          
        
        ,
      
    
    {\displaystyle {\mathcal {B}},}
  
 which by definition is the <a href="/facts/Intersection/hVW6QFWD">intersection</a> of all its elements:

ker
        ⁡
        
          
            B
          
        
         
        =
         
        
          ⋂
          
            B
            ∈
            
              
                B
              
            
          
        
        
        B
        .
      
    
    {\displaystyle \ker {\mathcal {B}}~=~\bigcap _{B\in {\mathcal {B}}}\,B.}
  
 
This definition is used in the theory of <a href="/facts/Filter_(set_theory)/7f3zk55u">filters</a> to classify them as being <a href="/facts/Free_filter/7f3zk55u">free</a> or <a href="/facts/Principal_filter/7f3zk55u">principal</a>. 
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Kernel (set theory) open-in-new

Kernel (set theory)