Translation of Integer Interval is Bijection

Theorem

Let $a, b, c \in \Z$ be integers.

Let $\closedint a b$ denote the integer interval between $a$ and $b$.

Then the mapping $T: \closedint a b \to \closedint {a + c} {b + c}$ defined as:

$\map T k = k + c$

is a bijection.

Proof

Note that if $k \in \closedint a b$, then indeed $k + c \in \closedint {a + c} {b + c}$.

Injectivity

Let $k, l \in \closedint a b$ with $k + c = l + c$.

By Integer Addition is Cancellable, $k = l$.

Thus $T$ is injective.

$\Box$

Surjectivity

Let $m \in \closedint {a + c} {b + c}$.

Then $m - c \in \closedint a b$.

Then $\map T {m - c} = m - c + c = m$.

Thus $T$ is surjective.

$\blacksquare$

Also see

• Cardinality of Integer Interval