Bounded and Unbounded Subsets of Real Numbers
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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.
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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
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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.
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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
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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.
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4.Greatest lower bound or infimum
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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.
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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.
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