Item 7 (01TN) — The Clowder Project

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

Descending Functions to Quotient Sets, IV. Let $R$ be an equivalence relation on $X$. If the conditions of Item 4 hold, then the following conditions are equivalent:
1. The map $\overline{f}$ is an injection.
2. We have $R=\mathrm{Ker}\webleft (f\webright )$.
3. For each $x,y\in X$, we have $x\sim _{R}y$ iff $f\webleft (x\webright )=f\webleft (y\webright )$.

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