Minimum
What is the Maximum Value?
The minimum value \(m\) of a set \(A\) is an element in \(A\) that is less than or equal to every other element in \(A\) $$ \begin{cases} m \in A \\ \\ m \le a \:\:\ \forall \:\: a \in A \end{cases} $$ The minimum value is often denoted as $$ m=\min(A) $$
An element can only be the minimum of a set if it belongs to the set itself.
If it does not belong to the set, it is referred to as a lower bound or infimum (Greatest Lower Bound).
Can a Set Be Without a Minimum? Yes, a set might not have a minimum value. Not all sets have a minimum value. For instance, the set of positive real numbers \(R^+\) does not have a maximum value because its domain is \(0,+\infty\). The number 0 is not part of the positive real numbers set. Between zero and any real number \(r\), there's always another real number \(r'\) within the interval \(0
A Practical Example
This set consists of 7 elements:
$$ A = \{ 1, 2, 4, -2, 6, -1, 3 \} $$
The minimum value of set \(A\) is -2
$$ \min(A) = -2 $$
because it is greater than or equal to all elements in the set
$$ -2 \le 1 \\ -2 \le 2 \\ -2 \le 4 \\ -2 \le -2 \\ -2 \le 6 \\ -2 \le -1 \\ -2 \le 3 $$
Uniqueness of the Minimum Value
If a set has a minimum value, that minimum value is unique.
Thus, two or more minimums cannot coexist within the same set.
However, it's possible for a set to be without a minimum.
Note. It's worth mentioning that a set cannot have duplicate elements within it. Therefore, if there is a minimum element, it is unique.
Proof
Assuming absurdly that a set has two minimum values
$$ m_1 \le a \:\: \forall a \in A $$
$$ m_2 \le a \:\: \forall a \in A $$
As minimums, both are elements of the set \(A\).
$$ m_1, m_2 \in A $$
Since each is less than or equal to every element in the set, \(M_1\) and \(M_2\) have a reciprocal order relation
$$ m_1 \le m_2 $$
$$ m_2 \le m_1 $$
By combining the two order relations, we obtain an equality relation
$$ ( m_1 \le m_2 ) \land (m_2 \le m_1) \Leftrightarrow m_1=m_2 $$
Therefore, the two minimums coincide and have the same value \(m_1 = m_2\).
This demonstrates the uniqueness of a set's minimum value.
And so on.
