nonempty set

[logistic_guy]

Show that every nonempty set of negative integers has a greatest element.

 

[khansaheb]

Show that every nonempty set of negative integers has a greatest element.

show us your effort/s to solve this problem.

 

[logistic_guy]

Let \(\displaystyle \text{A}\) be a nonempty set of negative integers.
And
Let \(\displaystyle \text{B} = \{-a \ | \ a \in \text{A} \}\)

Then,

\(\displaystyle \text{B}\) is a nonempty set of positive integers.

The well ordering theorem tells that set \(\displaystyle \text{B}\) must have a least element, let us call it \(\displaystyle b_0\).

Then,

\(\displaystyle b_0 = -a_0, \ \ \ a_0 \in \text{A}\).

Since \(\displaystyle b_0\) is the smallest element in set \(\displaystyle \text{B}\), we know that:

\(\displaystyle b_0 \leq -a\)

Or

\(\displaystyle -b_0 \geq a\)

Or

\(\displaystyle a_0 \geq a\)

This shows that \(\displaystyle a_0\) must be greater than or equal to every element in set \(\displaystyle \text{A}\).

Then,

\(\displaystyle a_0\) is the greatest element in set \(\displaystyle \text{A}\).