Universal closure no relation The empty set
P
        (
        c
        )
        →
         
        ∀
        
          x
        
        
          ∈
        
        
          X
        
        
        P
        (
        x
        )
        .
      
    
    {\displaystyle P(c)\to \ \forall {x}{\in }\mathbf {X} \,P(x).}
  The element c must be completely arbitrary; else, the logic does not follow: if c is not arbitrary, and is instead a specific element of the universe of discourse, then P(c) only implies an existential quantification of the propositional function.The empty set
By convention, the formula 
  
    
      
        ∀
        
          x
        
        
          ∈
        
        ∅
        
        P
        (
        x
        )
      
    
    {\displaystyle \forall {x}{\in }\emptyset \,P(x)}
   is always true, regardless of the formula P(x); see vacuous truth.Universal closure
The universal closure of a formula φ is the formula with no free variables obtained by adding a universal quantifier for every free variable in φ. For example, the universal closure of
no relation