The Well ordering principle states that
A least element exists in every non empty set of positive integers
Use the well Ordering principle to prove the following statement
' Any nonempty subset of negative integers has the greatest element '.
What I tried
By contradiction
I assume the statement
Any nonempty subset of negative integers has the least element to be true.
This means that the least element $a$ must be a negative integer but this contradicts with the Well ordering principle which states that the least element must be a positive integer. Hence the least element cannot be positive and negative at the same time which thus proves the orginal statement. Is my proof correct. Could anyone explain. Thanks
Best Answer
The negation of "Any nonempty subset of negative integers has the greatest element " is not "Any nonempty subset of negative integers has the least element", but rather "there exists a nonempty subset of negative integers that does not have a greatest element."
In general, the negation of a statement that has the universal quantifier ("for all ... ") has an existence quantifier ("there exists ...").
I think it's easier to prove this directly.
Hint: Let $A$ be a nonempty set of negative integers and consider the set $B=\{-a: a \in A\}$.
What can you say about the set $B$?