Bounds
Definitions
Definitions
For a given non-empty set of real numbers, S,
For a given non-empty set of real numbers, S,
A number M, is an Upper Bound if x ≤ M for all x ∈ S.
A number M, is an Upper Bound if x ≤ M for all x ∈ S.
A number m, is a Lower Bound if x ≥ m for all x ∈ S.
A number m, is a Lower Bound if x ≥ m for all x ∈ S.
A set is Bounded if an upper or lower bound exists.
A set is Bounded if an upper or lower bound exists.
A set is Unbounded if no upper or lower bound exists.
A set is Unbounded if no upper or lower bound exists.
The Maximum of S, max(S), is the element that is greater than all other elements of S.
The Maximum of S, max(S), is the element that is greater than all other elements of S.
The Minimum of S, min(S), is the element that is smaller than all other elements of S.
The Minimum of S, min(S), is the element that is smaller than all other elements of S.
The Supremum of S, sup(S), is the minimum of the set of all the Upper Bounds of S aka the Least Upper Bound (LUB)
The Supremum of S, sup(S), is the minimum of the set of all the Upper Bounds of S aka the Least Upper Bound (LUB)
The Infimum of S, sup(S), is the maximum of the set of all the Lower Bounds of S aka the Greatest Lower Bound (GLB)
The Infimum of S, sup(S), is the maximum of the set of all the Lower Bounds of S aka the Greatest Lower Bound (GLB)
The Completeness Axiom states that if a non-empty set of real number has an upper bound, it must have a Supremum. Similarly, if a non-empty set of real number has a lower bound, it must have a Infimum.
The Completeness Axiom states that if a non-empty set of real number has an upper bound, it must have a Supremum. Similarly, if a non-empty set of real number has a lower bound, it must have a Infimum.
Examples
Examples
A = {1, 2, 3, 5}, Max(A) = Sup(A) = 5, Min(A) = Inf(A) = 1
A = {1, 2, 3, 5}, Max(A) = Sup(A) = 5, Min(A) = Inf(A) = 1
B = {x : -2 ≤ x < 3}, Max(B) = Undefined, Sup(B) = 3, Min(B) = Inf(A) = -2
B = {x : -2 ≤ x < 3}, Max(B) = Undefined, Sup(B) = 3, Min(B) = Inf(A) = -2