This is Exercise II.6.11 of Bourbaki's Theory of sets (or II.6.10 in the newer edition).
Its intention is to generalize the correspondence between equivalence relations and partitions.
Given a symmetric and reflexive relation $R$ in a set $X$:
1) A connected component of $R$ is an equivalence class of the smallest equivalence relation containing $R$.
2) $R$ is said intransitive of order 1 iff for any distinct elements $x$, $y$, $z$, $w$ of $X$, if $xRy$, $xRz$, $xRw$, $yRz$, $yRw$, then $zRw$.
3) For distinct elements $a$ and $b$ of $X$ such that $aRb$, we define $C(a,b)=\{x\in X:(aRx\wedge bRx)\}$.
4) A constituent of $R$ is a set in the form $C(a,b)$ or a connected component that consists of a single element.
Then here are the questions:
a) Show that the intersection of two distinct constituents of $R$ has at most one element, and for distinct constituents $A$, $B$, $C$, at least one of $A\cap B$, $A\cap C$, $B\cap C$ is empty, or these three are equal. Moreover, the constituents are nonempty, and cover $X$, and for $x,y\in X$, then $xRy$ iff there is a constituent $A$ such that $\{x,y\}\subseteq A$.
b) Conversely, let $(U_i)_{i\in I}$ be a covering of $X$ by nonempty sets, such that for distinct indices $i$, $j$, then $U_i\cap U_j$ has at most one element, and for distinct indices $i$, $j$, $k$, at least one of $U_i\cap U_j$, $U_i\cap U_k$, $U_j\cap U_k$ is empty, or these three are equal. Let $R=\{\langle x,y\rangle\in X\times X:\exists i\in I:\{x,y\}\subseteq U_i\}$. Show that $R$ is symmetric and reflexive in $X$ and intransitive of order 1, and that the $U_i$ are the constituents of $R$.
This is how the exercise was written. I was able to do (a) and to prove that the relation in (b) is symmetric and reflexive in $X$ and intransitive of order 1. However, I think I found some error in (b), for I thought about the example $X=\{0,1,2\}$, with $U_0=\{0\}$, $U_1=\{0,1\}$, $U_2=\{0,2\}$, that satisfies the initial conditions of the question, but the set $U_0$ is not a constituent of $R$. I really do not know how to correct the exercise so that such relations and coverings be in correspondence, or, if any, what error I have done.
An even easier counterexample is $E = \{x, y\}$ where $x$ and $y$ are distinct, $X_{1} = \{x, y\}$ and $X_{2} = \{x\}$. Then $X_{2}$ is not a constituent of the relation $R$ for $E$.
To fix this, after "or these three are equal.", add the condition "For two distinct indices $\lambda$, $\mu$, $X_{\lambda}$ is not a subset of $X_{\mu}$."