Condition for Mapping from Quotient Set to be Surjection

Theorem
Let $S$ and $T$ be sets.

Let $\RR$ be an equivalence relation on $S$.

Let $f: S \to T$ be a mapping from $S$ to $T$.

Let $S / \RR$ be the quotient set of $S$ induced by $\RR$.

Let $q_\RR: S \to S / \RR$ be the quotient mapping induced by $\RR$. Let the mapping $\phi: S / \RR \to T$ defined as:
 * $\phi \circ q_\RR = f$

be well-defined.

Then:
 * $\phi$ is a surjection


 * $f$ is a surjection.
 * $f$ is a surjection.

Proof
We are given that:
 * $\phi \circ q_\RR = f$

is well-defined.

Note that from Quotient Mapping is Surjection, $q_\RR$ is a surjection.

Sufficient Condition
Let $\phi$ be a surjection.

Then from Composite of Surjections is Surjection:
 * $\phi \circ q_\RR$ is a surjection.

Necessary Condition
Let $f = \phi \circ q_\RR$ be a surjection.

Then from Surjection if Composite is Surjection:
 * $\phi$ is a surjection.

Hence the result.

Also see

 * Definition:Well-Defined Mapping


 * Condition for Mapping from Quotient Set to be Well-Defined
 * Mapping from Quotient Set when Defined is Unique
 * Condition for Mapping from Quotient Set to be Injection