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.
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.