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the relation R is partial order on the set Z where, R is defined as,  a R b  iff a=2b.

how to prove it.

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 A relation R on a set S is called a partial. ordering if it is reflexive, antisymmetric and transitive

1) reflexive:-  binary relation R over a set X is reflexive if ∀x ∈ X : x R x

 x R x  ie x=2x which is not satisfy

so not reflexive

2) anti symmetric :-R is anti-symmetric precisely if for all x and y in X. if x R y and y R x then x =y

x R y: x=2y

and y R x: y=2x

it does not mean that x=y 

so not antisymmetric

3) transtive:- if x R y and y R x then x R Z

x R y:x=2y

y R z: y=2z

then from above x=4z (but not x=2z for the condition x R z)

so not transitive

so not POSET

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if a relation is poset then it should be 

  • reflexive 
  • anti symmatric 
  • transative

but  reflexive and transative are fail here

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