Bounded and Unbounded Set

Unbounded in Mathematics means that the function is not bounded. You have already studied what a set is, and its basic properties, and you have also solved examples of sets.  So, the set which does not have a finite upper or lower bound is known as the Unbounded set, while the set which has a finite value of upper bound and lower bound is known as the bounded set. we will discuss the two types of sets, that is bounded and unbounded. Basically, You have studied that there are two types of sets that are finite and infinite.

Bounded Set –

Upper Bound of the Set

Consider a set A of real numbers,

Then, A is called bounded above if there is a number N so that any x A is less than or equal to N, N: x ⩽ N.

In this case, the number N is the UPPER BOUND OF SET.

Note:- If N is the upper bound for set A then any bigger number by N is also the upper bound of set A.

Conditions for any number to be the Upper bound of the set are given below:-

A number N belongs to a set is called the upper bound if;

1. N is the upper bound if any of the x belongs to A and satisfies the condition x ⩽ N.

x A and also satisfies x ⩽ N

2. If N is the upper bound for a set, then it is the smallest upper bound for that set. Any of the numbers which are smaller than N, will not be the upper bound of the set A.

3. Numbers greater than N can become the upper bound of set A. 

For example, You have an interval (4, 32]. 

The 32 is the least upper bound of the set, numbers greater than 32 can also become the upper bound.

Lower Bound of the set

Consider a set C of real numbers,

C is called bounded below if there is a number n so that any x C is greater than or equal to n, n : x n.

In this case, the number n is called the LOWER BOUND OF SET.

Note:- If n is the upper bound for set C then any smaller number by n is also the lower bound of set C.

Conditions for any number to be Lower bound of the set are given below:-

A number n belongs to a set is called the lower bound if;

1. n is lower bound if any of the x belongs to C and satisfies the condition x n.

x C and also satisfies x n

2. If n is the lower bound for a set, then it is the largest lower bound for that set. Any of the numbers which are greater than n, will not be the lower bound of the set C.

3. Numbers smaller than n can become the upper bound of set C. 

For example, You have an interval (4, 32]. 

The 4 is the greatest lower bound of the set, numbers greater than 4 cannot become the lower bound.

Conditions for Bounded Set

A set that has both upper and lower bounds that are set bounded above and bounded lower is known as a Bounded Set.

So for being a bounded set, there are two numbers, n and N for which n ≤ x ≤ N for any x ∈ A. If this condition satisfies set A, then A is called the Bounded Set.

Unbounded Set –

Unbounded means the things which are not bounded. In Mathematics, generally, the things which are not bounded are infinite. 

You have read that bounded sets should have both upper and lower bounds but, in contrast, Unbounded sets would not have an upper bound or lower bound. Both bounds can be infinite or if a single bound is infinite then it will also be called unbounded sets. 

There are some examples given below for unbounded sets:-

 (5,+∞), (−∞,45), (−2,+∞), (−∞,3), (−∞,+∞)

Also, the set of all real numbers and the set of all-natural numbers are also considered unbounded sets.

Properties of Bounded Set

1. The closure of a bounded set is always bounded.
2. Finite unions of bounded sets should be bounded sets.
3. Any subset of a bounded set is always a bounded set.

Difference between Bounded and Unbounded Set

The basic difference between a bounded and unbounded set is only the presence of finite and infinite numbers. 

If a set is having lower and upper bounds both or single then, it will be bounded otherwise it is an unbounded set. 

Conclusion

In this article, we have studied bounded and unbounded sets, their properties, and the basic difference between them. Bounded sets should have upper and lower bounds and unbounded sets do not have finite values. We have also gone through the properties of the bounded set. As we already know about the set and can solve problems associated with them, here we have taken the context of a set to describe the differences between the bounded set and unbounded set.