Problem understanding a question (picked from **Set Theory and Matrices** by I.Kaplansky)

Question

Suppose that a set X is expressed as a union of disjoint subsets. For $j, \, k \in X$ define $j \sim k$ that $j$ and $k$ lie in the same subset. Prove equivalency.

How do I start?

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My attempt:

I would write $X=\bigcup_{i}A_i$ [where $A_j \neq A_k$] [fault]

Then, I would let $B \subset X$. Therefore, if $k$ or $j$ $\in B$ then they would also have to lie within $X$. But hereafter, I am stuck...

Does anyone have any suggestions?

Answer 1

Here is how to start. First, write down the properties you need to show about $\sim$:

1. $\sim$ is reflexive: That is, $x \sim x$ for all $x \in X$.
2. $\sim$ is symmetric: That is, whenever $x_1,x_2 \in X$ satisfy $x_1 \sim x_2$, then also $x_2 \sim x_1$.
3. $\sim$ is transitive: That is, whenever $x_1,x_2,x_3 \in X$ satisfy $x_1 \sim x_2$ and $x_2 \sim x_3$, then also $x_1 \sim x_3$.

Now prove them. The first two are really just translations of the definition of $\sim$. The hardest is transitivity, and for it you need to use the fact that the sets in the partition given are not just disjoint but pairwise disjoint.