Infimum and Supremum of a Set (Need Help Finding Them)

In summary, the set B has a supremum of -3/2 and an infimum of -1. This is determined by plugging in values for n and noticing that the greatest and least values occur when n equals -2 and -1 respectively. Therefore, the supremum and infimum of B are -3/2 and -1 respectively.
  • #1
AutGuy98
20
0
Hey guys,

I have this Intermediate Analysis problem that I need help finding the answer to. This is what the question asks:

"Find the supremum and infimum of each of the following sets (considered as subsets of the real numbers). If a supremum or infimum doesn’t exist, then say so. No formal proof is necessary, but give a brief justification."

This is the set in question: B={(-1)^n+((-1)^n+1)/(2n)): n is a subset of Z (the set of integers) - {0}} (meaning "not including 0).

I started out by plugging in values for n from -5 to 5, not including 0, to see the answers produced, but I wasn't able to identify a pattern between any of them or anything like that. Not sure where to go from here with the problem, so any help you could give me would be helpful without a doubt. Thanks in advance.
 
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  • #2
AutGuy98 said:
Hey guys,

I have this Intermediate Analysis problem that I need help finding the answer to. This is what the question asks:

"Find the supremum and infimum of each of the following sets (considered as subsets of the real numbers). If a supremum or infimum doesn’t exist, then say so. No formal proof is necessary, but give a brief justification."

This is the set in question: B={(-1)^n+((-1)^n+1)/(2n)): n is a subset of Z (the set of integers) - {0}} (meaning "not including 0).

I started out by plugging in values for n from -5 to 5, not including 0, to see the answers produced, but I wasn't able to identify a pattern between any of them or anything like that. Not sure where to go from here with the problem, so any help you could give me would be helpful without a doubt. Thanks in advance.
Hi AutGuy, and welcome to MHB.

Let $x_n = (-1)^n + \dfrac{(-1)^{n+1}}{2n} = (-1)^n\left(1 - \dfrac1{2n}\right)$. Then $|x_n| = 1 - \dfrac1{2n}$.

If $n$ is positive then $|x_n|<1$ and if $n$ is negative then $|x_n|>1$. Also, if $n$ is small and negative then $|x_n|$ will be larger than if $n$ is large and negative.

In calculating $x_n$ for n from -5 to 5, you found (I hope) that the greatest and least values of $x_n$ occurred when $n=-2$ and $n=-1$.

From those hints, you should be able to "give a brief justification" of the fact that theose values are the sup and inf of the set $B$.
 
  • #3


Hi there,

It looks like you're trying to find the supremum and infimum of the set B. To do this, we need to first understand what these terms mean.

The supremum of a set is the smallest number that is greater than or equal to all the numbers in the set. In other words, it is the "least upper bound" of the set. The infimum, on the other hand, is the largest number that is less than or equal to all the numbers in the set. It is the "greatest lower bound" of the set.

Now, let's look at the set B. It is a set of numbers that are generated by plugging in different values for n. We can see that the terms inside the parentheses alternate between -1 and 1, and the terms outside the parentheses are all positive. When n is a positive even number, the term inside the parentheses is 1, and when n is a negative even number, the term is -1. Similarly, when n is a positive odd number, the term inside the parentheses is -1, and when n is a negative odd number, the term is 1.

Therefore, we can see that the terms in this set are always either 1 or -1, and they are alternating between these two values. This means that the supremum of the set would be 1, as it is the smallest number that is greater than or equal to all the numbers in the set. Similarly, the infimum would be -1, as it is the largest number that is less than or equal to all the numbers in the set.

To summarize, the supremum of the set B is 1, and the infimum is -1.

I hope this helps! Let me know if you have any other questions.
 

What is the definition of infimum and supremum of a set?

Infimum and supremum are two important concepts in mathematics that are used to describe the bounds of a set. The infimum of a set is the greatest lower bound, meaning it is the largest number that is less than or equal to all the numbers in the set. The supremum of a set is the least upper bound, meaning it is the smallest number that is greater than or equal to all the numbers in the set.

How are infimum and supremum related to minimum and maximum?

Infimum and supremum are related to minimum and maximum in that the infimum is the smallest number that is greater than or equal to all the numbers in the set, while the minimum is the smallest number in the set. Similarly, the supremum is the largest number that is less than or equal to all the numbers in the set, while the maximum is the largest number in the set.

What is the difference between infimum and minimum, and supremum and maximum?

The main difference between infimum and minimum is that infimum is a lower bound, while minimum is an actual element in the set. Similarly, supremum is an upper bound, while maximum is an actual element in the set. In other words, infimum and supremum may or may not be in the set, while minimum and maximum must be in the set.

How do you find the infimum and supremum of a set?

To find the infimum and supremum of a set, you first need to determine the lower and upper bounds of the set. Then, you can use mathematical techniques such as taking the minimum or maximum of the set, or using the limit of the set, to find the infimum and supremum. In some cases, the infimum and supremum may not exist for a set.

What are some real-life applications of infimum and supremum?

Infimum and supremum are used in various fields of mathematics, such as calculus, real analysis, and topology. They are also used in economics, finance, and optimization problems. In real life, they can be used to determine the minimum and maximum values of a function, to find the best possible solution to a problem, and to analyze data sets.

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