- #1
ar6
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Homework Statement
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The Attempt at a Solution
I am quite confused as to how to define the function. Any help would be appreciated.
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What's wrong with [itex]f(a) = \{ \, x \mid x \in A \wedge x \leq a \, \}[/itex]?ar6 said:I am quite confused as to how to define the function. Any help would be appreciated.
A partially ordered set, also known as a poset, is a set in which there is a binary relation that is reflexive, antisymmetric, and transitive. This means that for any two elements in the set, there is either a partial ordering between them or they are equivalent.
A partially ordered set does not have a defined ordering for all elements in the set, while a totally ordered set has a defined ordering for all elements. In a partially ordered set, there may be elements that are not comparable to each other, whereas in a totally ordered set, all elements can be compared.
In a partial ordering, elements may be equal to each other, while in a strict partial ordering, elements are not equal to each other. This means that in a strict partial ordering, there cannot be reflexive elements, whereas in a partial ordering, there can be reflexive elements.
Yes, a partially ordered set can have multiple minimal or maximal elements. This is because a partially ordered set only requires that there is a partial ordering between elements, not a total ordering. Therefore, there can be elements that are not comparable to each other and can all be considered minimal or maximal.
Some examples of partially ordered sets include the set of all subsets of a given set, where the relation is defined by subset inclusion. Another example is the set of all positive divisors of a given number, where the relation is defined by divisibility. Additionally, the set of all events in a given probability space can also be considered a partially ordered set, where the relation is defined by probability comparison.