Let $A$ be a nonempty set and $C$ is a partition of $A$.
A relation $\sim$ is defined as: $$For \ x, y \in A, x\sim y\ \ if \ and \ only \ if \ there \ exists \ U \in C \ such \ that\ x \in U and\ y \in U .$$
I have to prove that $\sim$ is an equivalence relation on the set $A$. Earlier in this problem, I had to prove that $\sim$ was an equivalence relation on $A = \{ a, b, c, d, e\}$ (it was part a of this 4 part problem). In this case, the set $A$ is being generalized. I'm confused on how to prove this without having specific "examples" that I could use like I did in part a of this question.
This follows directly by definition of partition. A partition of a set $A$, $C$ is a class of sets $U_\alpha$ so that every element of a $A$ is in some $U_\alpha$ and the $U_\alpha$ are disjoint.
Reflexive says:
There exists a $U$ so that $a\in U$ and $a \in U$.
By definition of partition there is some $U$ so that for each $a\in A$ there is a $U$ so that $a\in U$. (and therefore $a \in U$).
Symmetric says:
If there exists a $U$ so that $a,b \in U$ then there exists a $U$ so that $b,a \in U$.
(Don't need to say anything more.)
Transitive says:
If there exists a $U$ so that $a,b \in U$ and and $V$ so that $b,c\in V$ then there is a $W$ so that $a,c \in W$.
By definition of partition, the partitioning sets are disjoint. So if $b\in U$ and $b \in V$ then $U= V$. So if there exists a $U$ so that $a,b \in U$ and if there exists a $V$ so that $b,c \in V$ then $U=V$ and $a,b,c \in V$.