7. Let be a partial order set, and define a new relation S on A as...

  

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7. Let <A, R > be a partial order set, and define a new relation S on A as follows: Vx, y E A, xSyyRx, S is referred to as the dual relation of R. (1) Prove: S is also a partial order relation on A. (2) If R is the less than or equal to relation on the set of integers, then what kind of relation is S? If R is the divisibility relation on the set of positive integers, then what kind of relation is S? (3) What is the relationship between the maximal elements and minimal elements, and greatest element and least element in the partial order sets < A, R > and <A, S > ? 7. Let <A, R > be a partial order set, and define a new relation S on A as follows: Vx, y E A, xSyyRx, S is referred to as the dual relation of R. (1) Prove: S is also a partial order relation on A. (2) If R is the less than or equal to relation on the set of integers, then what kind of relation is S? If R is the divisibility relation on the set of positive integers, then what kind of relation is S? (3) What is the relationship between the maximal elements and minimal elements, and greatest element and least element in the partial order sets < A, R > and <A, S > ?