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Let $X$ be a nonempty set and let $R$ be an equivalence relation on $X.$ Given $X/R={[a]:a \in X}$. Prove that there is a map called the projection where $p_x:X\to X/R$ given by $p_x(t)=[t].$ Then this map is onto (surjective).

I know that by definition: Let $X$ be a nonempty set and let $R$ be an equivalence relation on $X$. The set of all equivalence classes, ${[a]:a\in X},$ is called the Quotient Set of $X$ modulo $R,$ written as $X/R.$

I am trying to teach this to myself. I spoke to a friend of mine who made me want to learn some of this. It seemed good to know. How would I prove that the projection map is onto given this nonempty set and equivalence relation. Can someone please help me? It would be nice if I can have a conversation with my friend showing him what I learned on this.

Andrés E. Caicedo
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mason
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    I wouldn't call this the projection map. Maybe the quotient map. – Asaf Karagila Dec 21 '14 at 22:59
  • @AsafKaragila Sorry. I was not sure what the proper name of this map would be. I did some research and they called it projection map in the question. Can you help me prove this? – mason Dec 21 '14 at 23:01
  • The projection map is from a Cartesian product onto one of the coordinates. This is a quotient map, since we "divide" by the equivalence relation, and it is somewhat like a quotient. As for helping, you have an answer. I would have left a short hint: use reflexivity of the relation. – Asaf Karagila Dec 21 '14 at 23:04

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Take any equivalence class (and denote it $\mathscr c$) of $X/R$ so by the definition of the equivalence relation $\mathscr c$ isn't empty. Let $a\in\mathscr c$ then we have

$$p_X(a)=\mathscr c$$ hence $p_X$ is onto.