# Definition:Ordering Induced by Injection

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## Definition

Let $\left({T, \le}\right)$ be an ordered set, and let $S$ be a set.

Let $f: S \to T$ be an injection.

Define $\le_f$ as the **ordering induced by $f$ on $S$** by:

- $\forall s_1, s_2 \in S: s_1 \le_f s_2 \iff f \left({s_1}\right) \le f \left({s_2}\right)$

That $\le_f$ is in fact an ordering is shown on Ordering Induced by Injection is Ordering.

## Total Ordering Induced by Injection

Let $\le$ be a total ordering.

Then $\le_f$ is also said to be the **total ordering induced by $f$ on $S$**.

This is appropriate by virtue of Injection Induces Total Ordering.

## Well-Ordering Induced by Injection

Let $\le$ be a well-ordering.

Then $\le_f$ is also said to be the **well-ordering induced by $f$ on $S$**.

This is appropriate by virtue of Injection Induces Well-Ordering.