The phrase "if every non-empty subset of A has a least element with respect to this relation" means that for any subset of set A that is not empty, there exists a minimum or smallest element within that subset according to a given relation.
To understand this concept better, let's break it down into smaller parts:
1. A non-empty subset: A subset is a set that contains elements from a larger set. In this context, a non-empty subset means that the subset has at least one element in it.
2. Least element: The least element refers to the smallest or minimum element within a set. It is the element that is smaller or equal to all other elements in the set.
3. Relation: A relation is a rule or condition that determines the order or comparison between elements. It can be based on numerical values, alphabetical order, or any other criteria.
So, when we say "if every non-empty subset of A has a least element with respect to this relation," we are stating that for any subset of set A that is not empty, there is always a minimum element in that subset based on a given relation. This means that you can always find the smallest element within any subset of A, regardless of the specific elements or the relation being used.
For example, let's consider set A = {3, 5, 7, 9} and the relation is "less than or equal to."
If we take the subset {3, 5}, the least element is 3.
If we take the subset {7, 9}, the least element is 7.
If we take the subset {3, 5, 7, 9}, the least element is 3.
In all cases, there is a least element within each subset of A based on the "less than or equal to" relation.
I hope this explanation clarifies the meaning of the phrase "if every non-empty subset of A has a least element with respect to this relation." Let me know if you have any further questions!