Define the relation R on the set Z by aRb if a-b is a non negative even integer. Show that R defines a partial order of Z. Is this partial order a total order? Answer with appropriate arguement.
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The relation R needs to be reflexive, antisymmetric and transitive, to be a partial order. All these conditions are obeyed by R as stated above. In fact, the popular notation to represent a partial order is <=, derived from precisely the same thing (a-b being a non-negative integer).
The relation R needs to be reflexive, antisymmetric and transitive, to be a partial order. All these conditions are obeyed by R as stated above. In fact, the popular notation to represent a partial order is <=, derived from precisely the same thing (a-b being a non-negative integer).Reflexive: aRa for all a in Z, since a-a=0, a non-negative integer, for any a.
The relation R needs to be reflexive, antisymmetric and transitive, to be a partial order. All these conditions are obeyed by R as stated above. In fact, the popular notation to represent a partial order is <=, derived from precisely the same thing (a-b being a non-negative integer).Reflexive: aRa for all a in Z, since a-a=0, a non-negative integer, for any a.Antisymmetric: If aRb and bRa, ie., if a-b>=0 and b-a>=0, then a>=b and b>=a, which is only possible if a=b.
The relation R needs to be reflexive, antisymmetric and transitive, to be a partial order. All these conditions are obeyed by R as stated above. In fact, the popular notation to represent a partial order is <=, derived from precisely the same thing (a-b being a non-negative integer).Reflexive: aRa for all a in Z, since a-a=0, a non-negative integer, for any a.Antisymmetric: If aRb and bRa, ie., if a-b>=0 and b-a>=0, then a>=b and b>=a, which is only possible if a=b.Transitive: If aRb and bRc, then a-b>=0 and b-c>=0, which gives a-c>=0 and hence aRc.
The relation R needs to be reflexive, antisymmetric and transitive, to be a partial order. All these conditions are obeyed by R as stated above. In fact, the popular notation to represent a partial order is <=, derived from precisely the same thing (a-b being a non-negative integer).Reflexive: aRa for all a in Z, since a-a=0, a non-negative integer, for any a.Antisymmetric: If aRb and bRa, ie., if a-b>=0 and b-a>=0, then a>=b and b>=a, which is only possible if a=b.Transitive: If aRb and bRc, then a-b>=0 and b-c>=0, which gives a-c>=0 and hence aRc.R is also a total order, because for any given a or b in Z, either aRb or bRa must be true (This is because either a<=b or b<=a).
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