Select all properties satisfied by reductions1 pointA ≤ A for all AFor all A, B, C, if A ≤ B and B ≤ C, then A ≤ C.For all A, B, C, if A ≤ B and A ≤ C, then B ≤ C.
Question
Select all properties satisfied by reductions1 pointA ≤ A for all AFor all A, B, C, if A ≤ B and B ≤ C, then A ≤ C.For all A, B, C, if A ≤ B and A ≤ C, then B ≤ C.
Solution
The properties satisfied by reductions are:
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Reflexivity: A ≤ A for all A. This means that any element is less than or equal to itself.
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Transitivity: For all A, B, C, if A ≤ B and B ≤ C, then A ≤ C. This means that if one element is less than or equal to a second element, and the second element is less than or equal to a third element, then the first element is less than or equal to the third element.
The third property, "For all A, B, C, if A ≤ B and A ≤ C, then B ≤ C", is not generally satisfied by reductions. This property is known as antisymmetry and it is not a property of reductions. It would imply that if A is less than or equal to both B and C, then B is less than or equal to C, which is not necessarily true in the context of reductions.
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