Bounded Set

Set is an important concept of Mathematics. It gives us logical knowledge of Mathematics. We are going to study the bounded set. In this article, we will discuss in detail the bounded set. 

Bounded Set

A set is called bounded if it has a finite size. If a set is consisting both upper and lower bound values, then we will consider it as a totally bounded set. Generally, A subset is used to cover the totally bounded set as it is having all elements of a universal set that is upper and lower bound. The numbers which are consisting of a bounded set come from the set of real numbers.

Bounded Set Definition in the Real Number

A set R of real numbers is known as Bounded set if there exists any real number t (it can be from outside set R) such that t r (for all r in R). 

Then the number t is called the upper bound of R. 

The terms bounded from below and lower bound will be similarly defined.

As a set R is having both upper and lower bounds then, it is contained in a finite interval.

Upper Bound and Lower Bound of the 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;

• 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

• 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.

• 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;

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

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.

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.

Examples of Bounded Set

Consider a set A having 10 natural numbers, as an example of a bounded set.

As, 

A = {1,2,3,4,5,6,7,8,9,10}

Interval for this set is [1,10]

The set P of prime numbers between 15 and 30.

P = {17, 19, 23, 29}

As this set contains a finite number of elements in this set, that’s why this set is also bounded.

Conclusion

In this article, we have studied bounded sets. Important conditions for bounded sets along with their examples. Bounded sets must have upper and lower bounds. If a set consists of a finite number of elements, then it is a bounded set. We have even discussed the conditions required for the upper bound of the set.