## 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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Graeme McRae.