Cardinality of Integer Interval
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 It has been suggested that this page or section be merged into I think there's something like this already, in the category concerned with the axiomatic construction of natural numbers, which came from Modern Algebra by Seth Warner. If appropriate, can be merged.. (Discuss) |
Theorem
Let $a, b \in \Z$ be integers.
Let $\left[{a \,.\,.\, b}\right]$ denote the integer interval between $a$ and $b$.
Then $\left[{a \,.\,.\, b}\right]$ is finite and its cardinality equals:
- $\begin{cases} b - a + 1 & : b \ge a - 1 \\ 0 & : b \le a - 1 \end{cases}$
Proof
Let $b < a$.
Then $\left[{a \,.\,.\, b}\right]$ is empty.
By Empty Set is Finite, $\left[{a \,.\,.\, b}\right]$ is finite.
By Cardinality of Empty Set, $\left[{a \,.\,.\, b}\right]$ has cardinality $0$.
Let $b \ge a$.
By Translation of Integer Interval is Bijection, there exists a bijection between $\left[{a \,.\,.\, b}\right]$ and $\left[{0 \,.\,.\, b - a}\right]$.
Thus $\left[{a \,.\,.\, b}\right]$ is finite of cardinality $b - a + 1$.
$\blacksquare$
