I just was thinking about the basic statement shown below relating equivalence relations and partitions. My question specifically is, how to pronounce this statement, $$ \overline a = \{x \in S: x \sim a \}. $$ Would it be correct to pronounce it as "the cell containing $a$ is defined as the set of all $x$ in $S$ such that $x$ is equivalent to a?" Is this correct pronunciation of $\overline a$ and $\sim$? As one of my math professors pronounces $\sim$ as "twiddle," the statement then might sound like, "the cell containing $a$ is defined as the set of all $x$ in $S$ such that $x$ twiddles a." Is that correct?
Let $S$ be a nonempty set and let $\sim$ be an equivalence relation on $S$. Then $\sim$ yields a partition of $S$, where $$ \overline a = \{x \in S: x \sim a \}. $$ Also, each partition of $S$ gives rise to an equivalence relation $\sim$ on $S$ where $a\sim b$ iff $a$ and $b$ are in the same cell of the partition.
Sure! It could also be read as, "the cell containing $a$ is defined as the set of all $x$ in $S$ such that $x$ relates to $a$." In particular cases, you would of course replace "$x$~$a$" with the given condition.