Upper and lower bounds

Imagine a group of numbers.

Some numbers are bigger than all others. We call these upper bounds.

Other numbers are smaller than all the rest. These are lower bounds.

A group can have many bounds. Some bounds are very tight.

We call a very tight upper bound a supremum. A tight lower bound is an infimum.

These bounds help us find limits. They show us where a group ends.

Imagine a group of numbers.

Some numbers are bigger than every number in that group. We call these upper bounds. For example, look at the set {1, 2, 3}. The number 5 is an upper bound. The number 10 is also an upper bound.

Other numbers are smaller than every number in the group. These are called lower bounds. In our set {1, 2, 3}, the number 0 is a lower bound. The number -5 is also a lower bound.

Some groups have many bounds. Some bounds are very tight. We call a very tight upper bound a supremum. This is the least upper bound. It is the smallest value that is still an upper bound. A tight lower bound is called an infimum. This is the greatest lower bound.

Bounds can also work with functions. A function is a rule that links inputs to outputs. One function can be an upper bound for another. This means its values are always higher or equal.

This helps us find the limits of how things change.

Imagine you have a group of numbers.

Some numbers are bigger than every single number in that group. We call these upper bounds. An upper bound is an element that is greater than or equal to every member of a subset. On the other hand, some numbers might be smaller than every number in the group. These are called lower bounds or minorants. A set that has an upper bound is said to be bounded from above. A set with a lower bound is called bounded from below.

How do these bounds work in real sets? Let us look at the set {1, 2, 3}. The number 5 is an upper bound for this set. The number 10 is also an upper bound. However, 0 is not a lower bound for {1, 2, 3} because it is not smaller than every element. In this set, the number 4 is an upper bound. Any number x where 3 is less than x works as an upper bound. The set {1, 2, 3} has both an upper bound and a lower bound.

Different kinds of number groups behave in different ways. Every finite subset of a non-empty totally ordered set has both bounds. The natural numbers always have a lower bound. This is because they have a least element like 0 or 1. An infinite subset of natural numbers cannot be bounded from above. An infinite subset of integers can be bounded from below or above. But an infinite subset of integers cannot be both.

Sometimes we look for the most perfect or tightest bounds. An upper bound is called a tight upper bound or a supremum. This is also known as the least upper bound. It means no smaller value can be an upper bound. A lower bound can also be tight. We call a tight lower bound an infimum. This is the greatest lower bound.

These terms help mathematicians find the exact limits of a group.

Bounds are not just for simple lists of numbers. We can also use them for functions. A function can be an upper bound for another function. This means its values are always higher or equal for every input. We call an upper bound sharp if equality holds for at least one value. This shows the constraint is optimal. We can also find an exact upper bound for certain types of products.

These ideas are very important in fields like PCF theory.

In order theory, mathematicians use specific terms to describe the limits of sets. These limits are known as upper and lower bounds. An upper bound, also called a majorant, is an element that is greater than or equal to every member of a subset. This concept helps us define the boundaries of a collection of items.

To understand the mechanism, imagine a subset within a preordered set. We look for an element that sits above every single part of that subset. If such an element exists, the set is described as being bounded from above or majorized. Conversely, we look for a lower bound, or minorant. A lower bound is an element that is less than or equal to every element in the subset. When a set has these limits, it is said to be bounded from below.

Different types of mathematical sets behave in very specific ways regarding these bounds. For example, consider the set {1, 2, 3}. The number 5 is an upper bound for this set because it is larger than 1, 2, and 3. However, 0 is not a lower bound for this set because it is not smaller than every element. In fact, any number x such that 3 is less than x acts as an upper bound. Any finite subset of a non-empty totally ordered set will always possess both upper and lower bounds.

Infinite sets follow more complex rules depending on their type. Every subset of the natural numbers has a lower bound. This is because natural numbers have a least element, which is either 0 or 1 depending on the convention used. However, an infinite subset of natural numbers cannot be bounded from above. Infinite subsets of integers are also restricted. An infinite subset of integers may be bounded from below or bounded from above, but it cannot be both.

Rational numbers offer even more variety in their behavior. An infinite subset of the rational numbers may or may not be bounded from below. Similarly, such a subset may or may not be bounded from above. This shows that the structure of the number system changes how limits apply. Mathematicians often search for the most precise limits possible.

We often look for the most efficient limits, which are called tight bounds. An upper bound is considered a tight upper bound if no smaller value can serve as an upper bound. This specific value is called a supremum, or the least upper bound. In the same way, a lower bound can be tight. A tight lower bound is called an infimum, or the greatest lower bound.

These concepts can also be applied to functions rather than just sets of numbers. If we have a function with a specific domain, an upper bound is another function. This upper bound must be greater than or equal to the original function for every value in the domain. An upper bound is called sharp if equality holds for at least one value. A sharp bound indicates that the constraint is optimal. This means the constraint cannot be reduced without breaking the inequality.

Finally, these ideas connect to very advanced areas of mathematics. An upper bound is an exact upper bound if every element that is strictly majorized by it is also majorized by some element of the subset. These exact upper bounds of reduced products of linear orders are very important. They play a significant role in a field known as PCF theory.