Maximum vs Maximal
It is often required by humans to denote the boundaries of things. If something cannot exceed beyond a certain limit, it is called maximum in the common sense. However, in the mathematical usage a much more rigorous definition has to be provided to prevent ambiguities.
Maximum
The greatest value of a set or a function is known as maximum. Consider the set {ai | i ∈ N}. The element ak where ak ≥ ai for all i is known as the maximum element of the set. If the set is ordered it becomes the last element of the set.
For example, take the set {1, 6, 9, 2, 4, 8, 3}. Considering all the elements 9 is greater than every other element in the set. Therefore, it is the maximum element of the set. By ordering the set, we get
{1, 2, 3, 4, 6, 8, 9}. In the ordered set, 9 (the maximum element) is the last element.
In a function, the largest element in the codomain is known as the maximum of the function. When a function reaches its maximum value the gradient becomes zero; i.e. its derivative at the maximum value is zero. This property is used to find the maximum value of functions. (You have to check the gradients of the curve on the sides of the point to confirm whether it is a maximum)
Maximal Element
Consider the set S, which is a subset of partially ordered set (A,≤). Then the element ak is said to be the maximal element if there is no element ai such that ak < ai. If ak is the greatest element of the partially ordered set, then it is unique. If it is not the greatest element, maximal element is not unique.
The concepts maximal is defined in the order theory and used in graph theory and many other fields.
What is the difference between Maximum and Maximal?
• Maximum is the greatest element of a set. When the set is ordered it becomes the last element of the set.
• Maximal is an element of a subset in a partially ordered set, such that there is no other element larger in the subset.
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