Projection (set theory)

<p>In <a href="/facts/Set_theory/1ptfsvUP">set theory</a>, a projection is one of two closely related types of <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> or operations, namely:
</p>
<ul><li>A <a href="/facts/Set_theory/1ptfsvUP">set-theoretic</a> operation typified by the 
  
    
      
        j
      
    
    {\displaystyle j}
  
th projection map, written 
  
    
      
        
          
            p
            r
            o
            j
          
          
            j
          
        
        ,
      
    
    {\displaystyle \mathrm {proj} _{j},}
  
 that takes an element 
  
    
      
        
          
            
              x
              →
            
          
        
        =
        (
        
          x
          
            1
          
        
        ,
         
        …
        ,
         
        
          x
          
            j
          
        
        ,
         
        …
        ,
         
        
          x
          
            k
          
        
        )
      
    
    {\displaystyle {\vec {x}}=(x_{1},\ \dots ,\ x_{j},\ \dots ,\ x_{k})}
  
 of the <a href="/facts/Cartesian_product/BfqBWzdL">Cartesian product</a> 
  
    
      
        (
        
          X
          
            1
          
        
        ×
        ⋯
        ×
        
          X
          
            j
          
        
        ×
        ⋯
        ×
        
          X
          
            k
          
        
        )
      
    
    {\displaystyle (X_{1}\times \cdots \times X_{j}\times \cdots \times X_{k})}
  
 to the value 
  
    
      
        
          
            p
            r
            o
            j
          
          
            j
          
        
        (
        
          
            
              x
              →
            
          
        
        )
        =
        
          x
          
            j
          
        
        .
      
    
    {\displaystyle \mathrm {proj} _{j}({\vec {x}})=x_{j}.}
  
</li>
<li>A function that sends an element 
  
    
      
        x
      
    
    {\displaystyle x}
  
 to its <a href="/facts/Equivalence_class/1d2sh0TB">equivalence class</a> under a specified <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a> 
  
    
      
        E
        ,
      
    
    {\displaystyle E,}
  
 or, equivalently, a <a href="/facts/Surjection/KezhLpCb">surjection</a> from a set to another set. The function from elements to equivalence classes is a surjection, and every surjection corresponds to an equivalence relation under which two elements are equivalent when they have the same image. The result of the mapping is written as 
  
    
      
        [
        x
        ]
      
    
    {\displaystyle [x]}
  
 when 
  
    
      
        E
      
    
    {\displaystyle E}
  
 is understood, or written as 
  
    
      
        [
        x
        
          ]
          
            E
          
        
      
    
    {\displaystyle [x]_{E}}
  
 when it is necessary to make 
  
    
      
        E
      
    
    {\displaystyle E}
  
 explicit.</li></ul>

Projection (set theory) open-in-new

Projection (set theory)