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Definition of Interval

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.

Internet references

Source: Wikipedia

Related pages in this website

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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