Understanding Supremum, Infimum, Maximum, Minimum, Upper and Lower Bounds

Mathematics, particularly in the context of sets and order theory, introduces several important concepts that help us understand the relationships between elements within a set. These concepts include supremum, infimum, maximum, and minimum, as well as upper and lower bounds. Understanding these terms is crucial for various applications in mathematics and related fields such as computer science, economics, and data analysis.

Partial Orders and Their Properties

To begin explaining these terms, let's first define what a partial order is. A partial order on a set ( P ) is a binary relation ( leq ) that satisfies the following properties:

Reflexive: For every element ( x ) in ( P ), ( x leq x ). Transitive: If ( x leq y ) and ( y leq z ), then ( x leq z ). Antisymmetric: If ( x leq y ) and ( y leq x ), then ( x y ).

Upper and Lower Bounds

Given a subset ( A ) of a partially ordered set ( P ), let's explore the concepts of upper and lower bounds, along with their related terms.

Upper Bounds

An element ( z ) in ( P ) is referred to as an upper bound of subset ( A ) if ( a leq z ) for every element ( a ) in ( A ).

Supremum (Least Upper Bound)

A unique element ( z ) in ( P ) is said to be the supremum of ( A ) if:

It is an upper bound of ( A ), and For every other upper bound ( u ) of ( A ), ( z leq u ).

The supremum is often called the least upper bound because it is the smallest element among all the upper bounds. It is noteworthy that the supremum itself might or might not be a member of ( A ).

Maximum

If an element ( z ) in ( A ) is both an upper bound and a member of ( A ), then ( z ) is referred to as the maximum element of ( A ). In this case, the maximum is explicitly a member of ( A ) and is also the supremum of ( A ).

Dual Concepts: Lower Bounds, Infimum, and Minimum

The concepts of lower bounds, infimum, and minimum can be defined in a dual manner by reversing the order of the inequalities involved:

Lower Bounds

An element ( z ) in ( P ) is referred to as a lower bound of subset ( A ) if ( z leq a ) for every element ( a ) in ( A ).

Infimum (Greatest Lower Bound)

A unique element ( z ) in ( P ) is said to be the infimum of ( A ) if:

It is a lower bound of ( A ), and For every other lower bound ( u ) of ( A ), ( u leq z ).

The infimum is often called the greatest lower bound because it is the largest element among all the lower bounds. Similarly, the infimum might or might not be a member of ( A ).

Minimum

If an element ( z ) in ( A ) is both a lower bound and a member of ( A ), then ( z ) is referred to as the minimum element of ( A ). In this case, the minimum is explicitly a member of ( A ) and is also the infimum of ( A ).

Examples and Practical Implications

Consider the set of real numbers less than 1. In this scenario:

The supremum is 1, which is not a member of the set (since it is not less than 1). The set has no maximum because every element in the set is strictly less than 1.

Now consider the set of real numbers less than or equal to 1:

The supremum is 1, which is a member of the set (since it is equal to 1). This 1 is also the maximum of the set.

For the set of real numbers greater than 1:

The infimum is 1, which is not a member of the set (since it is not greater than 1). The set has no minimum because every element in the set is strictly greater than 1.

Finally, for the set of real numbers greater than or equal to 1:

The infimum is 1, which is a member of the set (since it is equal to 1). The 1 is also the minimum of the set.

These examples help illustrate how supremum, infimum, maximum, minimum, upper, and lower bounds can vary based on the specific set and the elements it contains.

Conclusion

In summary, understanding the concepts of supremum, infimum, maximum, minimum, upper and lower bounds is essential for working with partially ordered sets. These terms provide a structured way to analyze and understand the relationships between elements within a set. Whether you are dealing with mathematical sets, functions, or other forms of data, these concepts can offer valuable insights and aid in problem-solving.

Further Reading

For more in-depth understanding of these concepts and their applications, you can explore the following resources:

Order Theory on Wikipedia Tagged Questions on Math Stack Exchange First Courses in Root Theory (and Other Books on Order Theory)