Let $a \in A \ and\ U \in C \ such\ that\ a \in U\ $. Prove that $[a] = U\ $.

Question

If you refer to this link and the question, it's this same question, this is the last part of the 4 part question .

The equivalence relation $\sim$ is defined as: $$\textsf{For }x,y\in A,x\sim y\textsf{ if and only if there exists }U\in C\textsf{ such that }x\in U\textsf{ and }y\in U$$

I have to prove that $[a] = U\ $ given that: $$a \in A\textsf{ and }U \in C\textsf{ such that }a \in U$$ I mentioned that the definition of a partition says that: $$\textsf{For all }a \in A\textsf{ there exists }U \in C\textsf{ such that }a \in U$$... but I'm not sure where to go from here.

Answer 1

This is a sequal to question Prove that $\sim$ is an equivalence relation on the set $A$

In that you prove that $x,y$ belonging to the same partitioning set is an equivalence relation.

Here you are asked to prove if $a\in$ the partitioning set $U$ that:

$[a] =\{x\in A| a\sim x\} = U$.

As $a \in U\subset A$ and that is distinct ($a$ is not in any other partitioning set) then

$[a] =\{x\in A| a\sim x\}= \{x\in A| a, x \in U\} = \{x\in A|x\in U\} = \{x \in U\} = U$.

That's all.