BOUNDED SETS

Bounded and Unbounded Subsets of Real Numbers

1. Upper bound

Def. Let S be a subset of real numbers. If there exists a real number u, such that

x ≤ u ∀ x ∈ R

then u is called an upper bound of S.

If there exists an upper bound for a set, then the set S is said to be bounded above.

Exp

1. S = { 1,2,3,4}

4 is an upper bound and all real number greater than 4 .

x ≤ 4 ∀ x ∈ S.

2. Z^+ = {1,2,3,4,5……………..}

Is not bounded above because no upper bound in this set.

3. Z^- = {………………1,2,3}

Is bounded above and 3 and greater than 3 is an upper bound of a set

2. Least upper bound or supremum:

Def. If s is an upper bound of a subset S of R and any Real number less than s is not an upper bound of S, then s is called the least upper bound (l.u.b) or Supremum(sup) of S.

3.Lower bound

Def. Let S be a subset of real numbers. If there exists a real number v, such that

x ≥ v ∀ x ∈ R.

then v is called a lower bound of S.

If there exists a lower bound for a set S, then the set S is said to be Bounded below.

Exp

1. S = { 1,2,3,4}

1 is a lower bound and all real number less than 1 .

x ≥ 1 ∀ x ∈ S.

2. Z^+ = {1,2,3,4,5……………..}

Is bounded below and 1 and less than 1 is lower bound in the set.

3. Z^- = {………………1,2,3}

Is not bounded above because no exists lower bound.

4.Greatest lower bound or infimum

1. If t is a lower bound of a subset S of R and any real number greater than t is not a lower bound of S, then t is called the greatest lower bound(g.l.b) or infimum (inf) of set S.

5.Bounded subset or unbounded subset.

A subset S of real numbers is said to be , if it is bounded above as well as bounded below.

v≤ x ≤ u.

A subset S of R is said to be if it is not bounded above or not bounded below.