A two-cell partition of a three-element set, the equivalence relation it induces listed pair by pair, and the quotient set recovered from it

Given: $u := \varnothing$, $v := \{\varnothing\}$, $w := \{\varnothing,\{\varnothing\}\}$, $A := \{u,v\} \cup \{w\}$, and $\sim$ as displayed.

reflexive: $a \sim a$ for every $a \in A$ (Equivalence relation, equivalence class, and the quotient set $A/{\sim}$).

symmetric: $a \sim b$ implies $b \sim a$, for all $a, b \in A$ (Equivalence relation, equivalence class, and the quotient set $A/{\sim}$).

transitive: $a \sim b$ and $b \sim c$ imply $a \sim c$, for all $a, b, c \in A$ (Equivalence relation, equivalence class, and the quotient set $A/{\sim}$).

$[a] \;:=\; \{\, b \in A \;:\; a \sim b \,\} \;\subseteq\; A$ (Equivalence relation, equivalence class, and the quotient set $A/{\sim}$).

$A/{\sim} \;:=\; \{\, [a] \;:\; a \in A \,\}$ (Equivalence relation, equivalence class, and the quotient set $A/{\sim}$).

The equivalence classes of an equivalence relation are nonempty, cover $A$, and are pairwise equal or disjoint (The equivalence classes of an equivalence relation are nonempty, cover $A$, and are pairwise equal or disjoint; conversely every such cover arises from exactly one equivalence relation).

$\{x,y\}$ is the set whose elements are exactly $x$ and $y$, and $\{x\} := \{x,x\}$ (The unordered pair $\{x,y\}$ and the singleton $\{x\} = \{x,x\}$).

There is exactly one set with no elements, written $\varnothing$ (There is exactly one set with no elements, written $\varnothing$).

$z \in a \cup b$ holds if and only if $z \in a$ or $z \in b$ ($\bigcup \varnothing = \varnothing$, $\bigcup \{a\} = a$, $\bigcup \{a,b\} = a \cup b$, $\bigcap \{a\} = a$, and $\bigcap \{a,b\} = a \cap b$).

$z \in A \times B$ holds if and only if $z = (a,b)$ for some $a \in A$ and some $b \in B$ (The Cartesian product $A \times B := \{\, z \in \mathcal{P}(\mathcal{P}(A \cup B)) : \exists a \in A\ \exists b \in B\ z = (a,b) \,\}$).

$(a,b) = (c,d)$ if and only if $a = c$ and $b = d$ ($(a,b) = (c,d)$ if and only if $a = c$ and $b = d$).

$(a,b) := \{\{a\},\{a,b\}\}$ (The Kuratowski ordered pair $(a,b) := \{\{a\},\{a,b\}\}$).

If every $z$ satisfies $z \in x$ if and only if $z \in y$, then $x = y$ (The Axiom of Extensionality: $\forall x\,\forall y\,(\forall z\,(z \in x \leftrightarrow z \in y) \to x = y)$).

A relation is a set $R$ every element of which is an ordered pair (Relation, $\operatorname{dom} R$, $\operatorname{ran} R$, $\operatorname{fld} R$, and the specialisations "relation from $A$ to $B$" and "relation on $A$").

$a \cup b := \bigcup\{a,b\}$ (The union $\bigcup x$ of a set, and the binary union $a \cup b := \bigcup \{a,b\}$).