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