Types of Bonded Set and their Significance

What is Bonded Set:

A set that is bounded above and below is called a bounded set. For instance, if S is a bounded set then there are two numbers, m and M so that m ≤ x ≤ M for any x ∈ S. It is sometimes convenient to lower m and/or increase M (if need be) and write |x| < C for all x ∈ S. The set which is not bounded is called an unbounded set. For instance, the interval (−2, 3) is bounded. Examples of unbounded sets: (−2, +∞),(−∞, 3), the set of all real numbers (−∞, +∞), and the set of all-natural numbers. 

Types of a bounded set are:

• Upper bounded set
• Lower bounded set

Bounds of sets of real numbers: 

Upper bounds of a set have the least upper bound (supremum): 

Let us Consider a set of real numbers S.

S is called bounded above if there is a number M such that any x ∈ S is less than, or equal to, M (x ≤ M). the value of M is called an upper bound for the set S.

Please note that if M is an upper bound for S then any bigger number than M is also an upper bound.

Not all sets have an upper bound, for example, the set of natural numbers does not.

The number B is called the least upper bound (or supremum) of the set S if: 

1) B is an upper bound: any x ∈ S satisfies x ≤ B, and 

2) B is the smallest upper bound, i.e., any smaller number is not an upper bound of t < B then there is x ∈ S with t < x

Notation:

B = sup S = sup x∈S x

Upper bounds of S may, or may not belong to S. 

For example, the interval (−2, 3) is bounded above by 100, 15, 4, 3.55, 3. The number 3 is its least upper bound. 

The interval (−2, 3] also has the number 3 as its least upper bound. When the supremum of S is a number that belongs to S, it is also called the maximum of S.

 Examples: 

1) The interval (−2, 3) has supremum equal to 3 and no maximum; (−2, 3] has supremum, and maximum, equal to 3. 

2) The function f(x) = x2 with domain [0, 4) has a supremum (equals 42 ), but not maximum. The function g(x) = x2 with domain [0, 4] has a maximum. It is equals g(4) = 42 .

The interval (−2, +∞) is not bounded above.

Bounded sets has a least upper bound:

This is a fundamental property of real numbers since it allows us to talk about limits.

 Theorem: Any non-empty set of real numbers which is bounded above has a supremum.

Proof.

We need a good notation for a real number given in its decimal representation.  The real number is in the form a = a₀a₁a₂a₃ a₄…    where a0 is an integer and a₁, a₂, a₃, … ∈ {0, 1, 2, …9}

 To find the real numbers, let us decide if the sequence of decimals ends up with nines: a = a₀a₁a₂a₂..an₉₉₉₉… (where an < 9) then we choose this number’s decimal representation as a = a0.a1 a2…(an + 1)0000…. (For instance, instead of 0.4999999.. we write 0.5.)

Let S be a non-empty set and the least upper bound of S. 

Consider first all the approximations by integers of the numbers a of S: if a = a₀a₁a₂… collect all the a0, it is a collection of integers. And It is bounded above. Then there is the largest one among them, call it B0.

Next, collect only the numbers in S which begin with B0. (There are some!) Call their collection S0. 

Any number in S \S₀ (number of S not in S₀) is smaller than any number in the S₀.

Look at the first decimal a1 of the numbers in S₀. Let B1 be the largest among them. Let S₁ be the set of all numbers in S₀ whose first decimal is B1.

Note that the numbers in S₁ begin with B₀.B₁ 

Also, note that any number in S \ S₁ is smaller than any number in S₁. Find the largest, B₂, etc.

Repeating the procedure we construct a sequence of smaller and smaller sets S₀, S₁, S₂, …S_n, … 

Note that every set Sn contains at least one element (it is not empty).

At each step n, we have constructed the set S n of numbers of S which start with B0.B1B2…Bn; the rest of the decimals can be anything. Also, all numbers in S \ Sn are smaller than all numbers of Sn. (The construction is by induction!) 

We end up with the number B = B₀.B₁ B₂…B_n B_n+1 …. 

We had to find B is the least upper bound. 

To show it is an upper bound, let an ∈ S. If a0 < B0 then a < B. Otherwise a0 = B0 and we go on to compare the first decimals. Either a1 < B1 therefore a < B or, otherwise, a1 = B1. Etc. So either a < B or a = B. So B is an upper bound. 

To show it is the least (upper bound), take any smaller number t < B. Then t differs from B at some first decimal, say at the nth decimal: t = B₀.B₁ B₂.…B_n−1t_nt_n+1 … and in < Bn. But then t is not in Sn and Sn contains numbers bigger than t. QED

Lower bounds 

By exchanging ”less than” < with ”greater than” > throughout section §3.1 we can similarly talk about lower bounds.

Here it is. 

S is called bounded below if there is a number m so that any x ∈ S is bigger than, or equal to m: x ≥ m. The number m is called a lower bound for the set S. 

Note that if m is a lower bound for S, then any smaller number is also lower. 

A number B is called the greatest lower bound (or infimum) of the set S if: 

1) b is a lower bound: any x ∈ S satisfies x ≥ B, and 

2) b is the greatest lower bound. In other words, any greater number is not a lower bound: 

if b < t then there is x ∈ S with x < t 

Notation: 

B = inf S = inf x ∈ S  x  

The greatest lower bounds of S may, or may not belong to S. For example, the interval (−2, 3) is bounded below by -100, -15, -4, and -2. −2 is its infimum (greatest lower bound). The interval [−2, 3) also has −2 as its infimum.

When the infimum of S belongs to S, it is said to be the minimum of S. 

Interval (−∞, −2) is not bounded below. If set S is not bounded below we write 

inf S = −∞ 

Theorem Any nonempty set of real numbers which is bounded below has an infimum. 

Proof. 

No, we need not repeat the proof of §3.2. We do as follows. Let S be a nonempty set that is bounded below. Construct the set T which contains all the opposites −a of the numbers an of S: 

T = {−a ; where a ∈ S} 

The set T is nonempty and is bounded above. By the Theorem of §3.2, T has a least upper bound, call it B. such that then its opposite, −B, is the greatest lower bound for S. Q.E.D.

Conclusion- 

In this article, we have discussed bonded sets. We have even discussed the bounds of a set of real numbers. The bounded set has the upper bounds and lower bounds which have been discussed in detail. The significance of bonded set has also been talked about in the article.