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 {m∈Am≤a ∀a∈A 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,+∞. 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≤1−2≤2−2≤4−2≤−2−2≤6−2≤−1−2≤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
m1≤a∀a∈A
m2≤a∀a∈A
As minimums, both are elements of the set A.
m1,m2∈A
Since each is less than or equal to every element in the set, M1 and M2 have a reciprocal order relation
m1≤m2
m2≤m1
By combining the two order relations, we obtain an equality relation
(m1≤m2)∧(m2≤m1)⇔m1=m2
Therefore, the two minimums coincide and have the same value m1=m2.
This demonstrates the uniqueness of a set's minimum value.
And so on.
