Supremum

Supremum is a principle in Mathematics and game theory which means “maximum upper bound” or “highest card”. In other words, the value of an ordered pair 

with the maximum value for each coordinate.  In general, the principles of Supremum are related to maximising profits, minimising risks, and playing poker. It has been shown that it can also be used in protein folding as well as many other mathematical applications. Supremum notes will help you in understanding Mathematics and game theory. The mathematical concept of supremum is the concept of maximum upper bound or highest card. Supremum study material will provide more insight on supremum.

What is Supremum?

Supremum is a concept of set theory in Mathematics that refers to the highest element of a set that has the property that it cannot be greater than itself. For example, the supremum for positive integers would be positive integer one.

Since many mathematical operations can only be performed on sets of values, Supremum can sometimes provide very important insights into how figures relate to each other in Maths. A definition of these operations will also be given.

Supremum in Set Theory: 

In order to define what constitutes a “set,” we must first define what constitutes “a set.” A set is an abstract collection without any specific pattern or order to it and which consists of many item(s). If the elements in a set are all distinct, then we can say that the set is finite. If every element in a set is equal to another (and therefore not a subset of another) we have said that the elements are all mutually equinumerous. If we add one or more new element(s) to an already existing finite set, and the new element is included in at least 1 out of all the remaining members, then this new addition constitutes a “subset” of the original set.

Definitions and Examples:

Definition: This is the definition which gives a comprehensible description of supremum: 

“In a set or multiset, the supremum (or maximum) over all finite subsets A, B, C, …of some property P is the largest value of P over all members of those sets. It is related to cardinality by replacing finite with infinite.

Example:

In this example we have the set of the numbers 2 and 7 with the property of being odd. Between these two numbers is infinity, so the supremum is infinity.

Example:

Another example would be in playing poker:

Let us say that there are six players in a poker game, and all are at even money with each other. The pot will contain $100, and each player has agreed to put up $100 at the start of each hand. For simplicity’s sake we will say that every player has five cards before they make their wager, and that every player has one card face down (this may not be exactly how poker actually works). The first player will have five cards, four of which are the same card, and one of which is a different card. The next player will have four cards the same as the first player and one different, and this pattern continues until we get to the sixth player who has four cards all the same as each other and one different. Each player will have to use their best judgement to determine what they believe their hand is worth at even money. This means that each player must raise or fold based on mathematical calculations of how much profit they can make from their hand with this additional information, since everyone knows everyone else’s hands already. This model or idea can be expanded for more players than just six.

Using Supremum:

The most useful application of supremum is in the area of risk analysis and insurance. Many large insurance companies are using this concept to determine the size of their risk in order to determine if they need to raise their premiums. Since many insurance companies calculate risk it is important for an individual to understand how one insurance company can make a profit on a given level of risk. The basic method used is as follows: first, determine the highest value that could possibly be obtained from any set (for any number of people), then multiply this by a factor (such as 1 million) which would be the total loss that could possibly occur in a year.

How does Supremum relate to other concepts?

Concept of infinity: The concept of infinity and supremum can be related in a few different ways. One way is that supremum is the greatest thing possible, while infinity is a number which is greater than any other (including itself). 

Properties of infinities: In some ways it could be said that there are infinitely many properties in the world. We could also say that there are two infinities, or two supreme sets. 

How Supremum is different from Infimum?

Supremum is the concept of greatest which is different from the concept of Infimum. If a number is smaller than any other number then it is said to be Infimum or less than. An infimum must be between its two opposites, but in a sense an infimum can also be an outlier to itself. The outlier property would make it different from supremum, and this would make them equal in certain instances. For example, in the example above stated, if the values of infinity and one were both infinities then they would both be supremum because they are equal to each other.

Conclusion:

Supremum can also be called a maximum. We know that Supremum is the greatest or highest among something, so there will be no maximum to anything. The common thing which we do with supremum is that we maximise some particular thing in our life by putting lots of effort and doing harder work. By doing this process, we can accomplish the principle which is generally known as the suprema principle.