An Interval is a subset S of a totally
ordered set T with the property that whenever
- x and y are in S and
- z is in T and
- x < z < y
then z is in S.
A "totally ordered" set is a set over which the ≤ operator is reflexive, antisymmetric, transitive, and total...
reflexive: a ≤ a
antisymmetric: if a ≤ b and b ≤ a then a = b
transitive: if a ≤ b and b ≤ c then a ≤ c
total: a ≤ b or b ≤ a
The reflexive property restricts the meaning of the ≤ operator, and the other three properties are satisfied by the real numbers.
Go back to Calculus Home or visit Calculus Theorems
Subdividing an Interval
Arithmetic Rules -- properties of equality, addition, multiplication, and in particular the Axioms of Real Arithmetic, including the completeness axiom.
Upper Bound -- definition of "upper bound" and "least upper bound" of either sets or functions
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