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Homework Statement
State the definitions of greatest and least elements in terms of strict orderings.
Homework Equations
Let [tex]\leq[/tex] be an ordering of [tex]A[/tex] and [tex]<[/tex] be a strict ordering on [tex]A[/tex], and let [tex]B \subseteq A[/tex].
[tex]b \in B[/tex] is the greatest element of [tex]B[/tex] in the ordering [tex]\leq[/tex] if, for every [tex]x \in B[/tex], [tex]x \leq b[/tex].
[tex]b \in B[/tex] is the least element of [tex]B[/tex] in the ordering [tex]\leq[/tex] if, for every [tex]x \in B[/tex], [tex]b \leq x[/tex].
The Attempt at a Solution
[tex]b \in B[/tex] is the greatest element of [tex]B[/tex] in the ordering [tex]<[/tex] if, for every [tex]x \in B[/tex], [tex]x < b[/tex].
But then there is no greatest element, because [tex]x < b[/tex] implies [tex]x \neq b[/tex]. So maybe it should be:
[tex]b \in B[/tex] is the greatest element of [tex]B[/tex] in the ordering [tex]<[/tex] if, for every [tex]x \in B[/tex] and [tex]x \neq b[/tex], [tex]x < b[/tex]?
[tex]b \in B[/tex] is the least element of [tex]B[/tex] in the ordering [tex]<[/tex] if, for every [tex]x \in B[/tex], [tex]b < x[/tex].
But then there is no least element, because [tex]x < b[/tex] implies [tex]x \neq b[/tex]. So maybe it should be:
[tex]b \in B[/tex] is the least element of [tex]B[/tex] in the ordering [tex]<[/tex] if, for every [tex]x \in B[/tex] and [tex]x \neq b[/tex], [tex]b < x[/tex]?
While I am at it I might as well also check that I've got some other definitions right:
[tex]b \in B[/tex] is a maximal element of [tex]B[/tex] in the ordering [tex]<[/tex] if there exists no [tex]x \in B[/tex] such that [tex]b < x[/tex].
[tex]b \in B[/tex] is a minimal element of [tex]B[/tex] in the ordering [tex]<[/tex] if there exists no [tex]x \in B[/tex] such that [tex]x < b[/tex].
[tex]a \in A[/tex] is an upper bound of [tex]B[/tex] in the ordered set [tex](A, <)[/tex] if [tex]x < a[/tex] for all [tex]a \in B[/tex].
[tex]a \in A[/tex] is called a supremum of [tex]B[/tex] in [tex](A, <)[/tex] if it is the least element of the set of all upper bounds of [tex]B[/tex] in [tex](A, <)[/tex].
[tex]a \in A[/tex] is a lower bound of [tex]B[/tex] in the ordered set [tex](A, <)[/tex] if [tex]a < x[/tex] for all [tex]x \in B[/tex].
[tex]a \in A[/tex] is called an infimum of [tex]B[/tex] in [tex](A, <)[/tex] if it is the greatest element of the set of all lower bounds of [tex]B[/tex] in [tex](A, <)[/tex].
Let [tex]a, b \in A[/tex], and let [tex]<[/tex] be an ordering of [tex]A[/tex]. We say that [tex]a[/tex] and [tex]b[/tex] are comparable in the ordering [tex]<[/tex] if [tex]a < b[/tex] or [tex]b < a[/tex]. We say that [tex]a[/tex] and [tex]b[/tex] are incomparable if they are not comparable (i.e., if [tex]a \neq b[/tex] and neither [tex]a < b[/tex] nor [tex]b < a[/tex] .