6.3. Let (A,
Given: and are partially ordered sets.
A Cartesian product A x B is defined with relation by setting if and only if and .
As and are partially ordered sets. Therefore, and are reflexive, antisymmetric and transitive.
consider an element of A x B.
(i) As A and B are posets, therefore, and .
Therefore, .
Thus the relation is reflexive relation on A x B.
(ii) Now consider , A and B are antisymmetric.
Therefore, implies and implies that .
Hence the relation is antisymmetric relation on A x B.
(iii) If and are defined on A x B.
As A and B are transitive, therefore,
and implies that .
Also and implies that .
That is and implies that .
Thus is transitive relation on A x B.
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