Maximum

What is the Maximum Value?

The maximum M of a set A is an element of A that is greater than or equal to all other elements in set A $$ \begin{cases} M \in A \\ \\ M \ge a \:\:\ \forall \:\: a \in A \end{cases} $$ The maximum value is often denoted as $$ M=max(A) $$

An element can only be the maximum of a set if it belongs to the set itself.

If it does not belong to the set, it is called an upper bound or Supremum (Least Upper Bound).

Can a set be without a maximum? Yes, a set might not have a maximum value. Not all sets have a maximum value. For example, the set of real numbers R does not have a maximum value because its domain is (-∞,+∞). The symbol +∞ is not a maximum value.

A Practical Example

The set A consists of 7 elements

$$ A = \{ -1, 0, 4, 2, 6, 1, 3 \} $$

The maximum value of set A is 6

$$ max(A) = 6 $$

because it is greater than or equal to all elements in the set

$$ 6 \ge -1 \\ 6 \ge 0 \\ 6 \ge 4 \\ 6 \ge 2 \\ 6 \ge 6 \\ 6 \ge 1 \\ 6 \ge 3 $$

Uniqueness of the Maximum Value

If a set has a maximum value, that maximum value is unique.

Therefore, there cannot be two or more maximums in a set.

A set can, however, be without a maximum.

Note: It's important to remember that a set cannot have duplicate elements within it. Thus, if there is a maximum element in the set, this element is unique.

Proof

Assume, for the sake of contradiction, that a set has two maximum values

$$ M_1 \ge a \:\: \forall a \in A $$

$$ M_2 \ge a \:\: \forall a \in A $$

As maximums, both values are elements of set A.

$$ M_1, M_2 \in A $$

Since each is greater than or equal to all elements in the set, M₁ and M₂ must be equal to each other

$$ M_1 \ge M_2 $$

$$ M_2 \ge M_1 $$

Combining these order relations leads to an equality

$$ ( M_1 \ge M_2 ) ∧ ( M_2 \ge M_1 ) \Leftrightarrow M_1=M_2 $$

Therefore, the two maximums coincide and have the same value M₁ = M₂.

This proves the uniqueness of the maximum value in a set.

And so on.