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bedi
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Let X be an initial segment of a set A. By definition, if x is in X, a is in A and x>a then a is in X too. Can we say that some elements of A that are greater than x are also in X? Or X only consists of elements smaller than x?
No, that's why the word "initial" is used. If X is an initial segment of A and x is a specific member, then that definition says that all members of A less than x are in X. It does NOT say anything about numbers larger than x, one way or the other.bedi said:Let X be an initial segment of a set A. By definition, if x is in X, a is in A and x>a then a is in X too. Can we say that some elements of A that are greater than x are also in X? Or X only consists of elements smaller than x?
An initial segment is a subset of a partially ordered set in which any two elements in the subset are comparable and any element that is less than another element in the subset is also in the subset.
Initial Segment X refers to the subset of elements in a partially ordered set that are less than or equal to the element X. It is also known as a lower set or a down-set.
The inclusion of elements >x in an initial segment is determined by the partial order of the set. If there are elements that are greater than x in the set, they will not be included in the initial segment. If there are no elements greater than x, then the initial segment will include all elements in the set.
Determining if elements >x are included in an initial segment can help in understanding the structure and properties of the partially ordered set. It can also be useful in proving theorems and solving problems related to the set.
Yes, elements >x can still be included in an initial segment even if the partial order is not strict. This is because the inclusion of elements in an initial segment is determined by the order relation, not the strictness of the order.